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assyrianic/bool_algebra_overview.txt

Last active January 23, 2024 18:59
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bool_algebra_overview.txt
Annulment rule: `a + 1 = 1`.
Idempotent rule: `a + a = a`, `aa = a`.
Complement rule: `aa̅ = 0`.
Absorption rule: `a + ab == a`, `a + a̅b == a+b`.
Identity rule: `a + 0 = a`, `a + a̅ = 1`, `a1 = a`.
Commutative rule: `ab == ba`, `a+b == b+a`, `abc == cba`.
Distributive rule: `a+(bc) == (a+b)(a+c)`, `a(b+c) == ab + ac`, `a(a+b+c) == aa + ab + ac => a + ab + ac => a + ac => a`.
Associative rule: `(a + b) + c == a + (b + c) == a + b + c`, `(ab)c == a(bc) == abc`.
Consensus/Resolvent rule: `ab + a̅c + bc == ab + a̅c`, `(a+b)(a̅+c)(b+c) == (a+b)(a̅+c)`
Proof:
𝒇 = ab + a̅c + bc 𝒇 = ab + a̅c + bc
𝒇 = ab + a̅c + bc1 𝒇 = (abc + abc̅) + (a̅bc + a̅b̄c) + (abc + a̅bc)
𝒇 = ab + a̅c + bc(a + a̅) 𝒇 = ab(c + c̅) + a̅c(b + b̄) + 1
𝒇 = ab + a̅c + abc + a̅bc 𝒇 = ab + a̅c
𝒇 = ab(1 + c) + a̅c(1 + b)
𝒇 = ab + a̅c
treat NOT as their own variable. a̅b̅c̅
De Morgan's Theorem:
a̅ + b̅ => ā∙̄b̄ NAND
a̅b̅ => ā+̄b̄ NOR
a̅b + ab̅ => a⨁b XOR
a̅b̅ + ab => a̅⨁̅b̅ => a⨀b XNOR
NAND Logic:
ab = !!(ab) --> AND
a+b = !(!a!b) --> OR
a = !(a & a) --> NOT
a⨁b = !( ![a & !b] & ![!a & b] ) --> XOR
a⨀b = !( ![!a & !b] & ![a & b] ) --> XNOR
NOR Logic:
a = !(a+a) --> NOT
a+b = !!(a+b) --> OR
ab = !(!a + !b) --> AND
̅
Distributive Example:
(a + b)(a + c) => Do FOIL like in regular algebra.
= aa + ac + ba + bc => Factor out.
= a(1 + c + b) + bc => Resolve parentheses, remember Identity rule!
= a(1) + bc => Identity Rule again.
= a + bc => Done!
Simplification Example:
ac̅ + abc̅ => Factor out.
= ac̅(1 + b) => Identity Rule in parentheses
= ac̅ => Done!
Full Adder Descriptor:
-Sum: a̅b̅c + a̅bc̅ + ab̅c̅ + abc => Factor out like terms.
= c̅(a̅b + ab̅) + c(a̅b̅ + ab) => Simplify & De Morgan's theorems.
= c̅(a⨁b) + c(a̅⨁̅b̅) => Re-apply De Morgan's theorem.
= a⨁b⨁c => Done.
-Carry out: a̅bc + ab̅c + abc̅ + abc => Factor out like terms.
= a̅bc + ab̅c + ab(c̅ + c) => Simplify.
= a̅bc + ab̅c + ab(1) => Simplify.
= c(a̅b + ab̅) + ab => Factor.
= c(a⨁b) + ab => Done.
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc
= a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc => search for like terms
= a̅b̅c + a̅b(c̅ + c) + ab̅c̅ + ab̅c + abc̅ + abc => factor out
= a̅b̅c + a̅b + ab̅c̅ + ab̅c + abc̅ + abc => search again for like terms
= a̅b̅c + a̅b + ab̅(c̅ + c) + abc̅ + abc => factor out again
= a̅b̅c + a̅b + ab̅ + abc̅ + abc => search again for like terms
= a̅b̅c + a̅b + ab̅ + ab(c̅ + c) => factor out again
= a̅b̅c + a̅b + ab̅ + ab => search again for like terms
= a̅b̅c + a̅b + a(b̅ + b) => factor out again
= a̅b̅c + a̅b + a => search again for like terms
= a̅(b̅c + b) + a => factor out again & distributive law
= a̅(c + b) + a => distributive law again `a+(bc) == (a+b)(a+c)`
= (a̅ + a)(a + b + c) => distributive law applied
= a + b + c => done | 3-input OR
Max Value: 16 | Total bits: 4 | Number of Vars: 2
𝒇 = 0 [DONE]
𝒇 = ab [DONE]
𝒇 = ab̅ [DONE]
𝒇 = ab̅ + ab => a(b̅ + b) => a [DONE]
𝒇 = a̅b [DONE]
𝒇 = a̅b + ab => b(a̅ + a) => b [DONE]
𝒇 = a̅b + ab̅ => a⨁b XOR => a != b [DONE]
𝒇 = a̅b + ab̅ + ab => a̅b + a(b̅ + b) => a̅b + a => (a + a̅)(a + b) => 1(a + b) => a + b [DONE]
𝒇 = a̅b̅ => ā+̄b̄ NOR [DONE]
𝒇 = a̅b̅ + ab => a̅⨁̅b̅ == a⨀b => a==b XNOR [DONE]
𝒇 = a̅b̅ + ab̅ => b̅(a̅ + a) => b̅ [DONE]
𝒇 = a̅b̅ + ab̅ + ab => a̅b̅ + a(b̅ + b) => a̅b̅ + a => (a + a̅)(a + b̅) => 1(a + b̅) => a + b̅ [DONE]
𝒇 = a̅b̅ + a̅b => a̅(b̅ + b) => a̅ [DONE]
𝒇 = a̅b̅ + a̅b + ab => a̅(b̅ + b) + ab => a̅ + ab => (a̅ + a)(a̅ + b) => 1(a̅ + b) => a̅ + b [DONE]
𝒇 = a̅b̅ + a̅b + ab̅ => a̅(b̅ + b) + ab̅ => a̅ + ab̅ => (a̅ + a)(a̅ + b̅) => 1(a̅ + b̅) => a̅ + b̅ => ā∙̄b̄ NAND [DONE]
𝒇 = 1 [DONE]
Max Value: 256 | Total Bits: 8
𝒇 = 0 [DONE]
𝒇 = abc [DONE]
𝒇 = abc̅ [DONE]
𝒇 = abc̅ + abc => ab(c̅ + c) => ab [DONE]
𝒇 = ab̅c [DONE]
𝒇 = ab̅c + abc => ac(b̅ + b) => ac [DONE]
𝒇 = ab̅c + abc̅ => a(b̅c + bc̅) => a(b⨁c) [DONE]
𝒇 = ab̅c + abc̅ + abc => ab̅c + ab [DONE]
𝒇 = ab̅c̅ [DONE]
𝒇 = ab̅c̅ + abc => a(b⨀c) [DONE]
𝒇 = ab̅c̅ + abc̅ => ac̅(b̅ + b) => ac̅ [DONE]
𝒇 = ab̅c̅ + abc̅ + abc => ab̅c̅ + ab(c̅ + c) => ab̅c̅ + ab => a(b̅c̅ + b) => a(b + c̅) => ab + ac̅ [DONE]
𝒇 = ab̅c̅ + ab̅c => ab̅ [DONE]
𝒇 = ab̅c̅ + ab̅c + abc => ab̅(c̅ + c) + abc => a(b̅ + bc) => a(b̅ + c) => ab̅ + ac [DONE]
𝒇 = ab̅c̅ + ab̅c + abc̅ => ab̅(c̅ + c) + abc̅ => ab̅ + abc̅ => a(b̅ + bc̅) => a(b̅ + c̅) => ab̅ + ac̅ [DONE]
𝒇 = ab̅c̅ + ab̅c + abc̅ + abc => ab̅(c̅ + c) + ab(c̅ + c) => ab̅ + ab => a(b̅ + b) => a [DONE]
𝒇 = a̅bc [DONE]
𝒇 = a̅bc + abc => bc(a̅ + a) => bc [DONE]
𝒇 = a̅bc + abc̅ => b(a̅c + ac̅) => b(a⨁c) [DONE]
𝒇 = a̅bc + abc̅ + abc => bc(a̅ + a) + abc̅ => bc + abc̅ => b(c + ac̅) => b(c + a) => ab + bc [DONE]
𝒇 = a̅bc + ab̅c => c(a̅b + ab̅) => c(a⨁b) [DONE]
𝒇 = a̅bc + ab̅c + abc => ab̅c + bc(a̅ + a) => ab̅c + bc => c(b + ab̅) => c(a + b) => bc + ac [DONE]
𝒇 = a̅bc + ab̅c + abc̅ [DONE]
𝒇 = a̅bc + ab̅c + abc̅ + abc => a̅bc + ab̅c + ab(c̅ + c) => a̅bc + ab̅c + ab => a̅bc + a(b̅c + b) => a̅bc + ab + ac => b(a + a̅c) + ac => b(a + c) + ac => ab + bc + ac [DONE]
𝒇 = a̅bc + ab̅c̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + abc => bc(a̅ + a) + ab̅c̅ => bc + ab̅c̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + abc̅ => a̅bc + ac̅(b̅ + b) => a̅bc + ac̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + abc̅ + abc => bc(a̅ + a) + ac̅(b̅ + b) => bc + ac̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + ab̅c => a̅bc + ab̅(c̅ + c) => a̅bc + ab̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + ab̅c + abc => bc(a̅ + a) + ab̅(c̅ + c) => bc + ab̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅bc + ab̅(c̅ + c) + abc̅ => a̅bc + ab̅ + abc̅ => a̅bc + a(b̅ + bc̅) => a̅bc + a(b̅ + c̅) => a̅bc + ab̅ + ac̅ [DONE]
𝒇 = a̅bc + ab̅c̅ + ab̅c + abc̅ + abc => a̅bc + ab̅(c̅ + c) + ab(c̅+c) => a̅bc + ab̅ + ab => ab̅ + b(a + c) => ab̅ + ab + bc => a(b̅ + b) + bc => a + bc [DONE]
𝒇 = a̅bc̅ [DONE]
𝒇 = a̅bc̅ + abc => b(a̅c̅ + ac) => b(a⨀c) [DONE]
𝒇 = a̅bc̅ + abc̅ => bc̅(a̅ + a) => bc̅ [DONE]
𝒇 = a̅bc̅ + abc̅ + abc => a̅bc̅ + ab(c̅ + c) => a̅bc̅ + ab => b(a + a̅c̅) => b(a + c̅) => ab + bc̅ [DONE]
𝒇 = a̅bc̅ + ab̅c [DONE]
𝒇 = a̅bc̅ + ab̅c + abc => a̅bc̅ + ac(b̅ + b) => a̅bc̅ + ac [DONE]
𝒇 = a̅bc̅ + ab̅c + abc̅ => bc̅(a̅ + a) + ab̅c => bc̅ + ab̅c [DONE]
𝒇 = a̅bc̅ + ab̅c + abc̅ + abc => a̅bc̅ + abc̅ + ac(b̅ + b) => bc̅(a̅ + a) + ac => bc̅ + ac [DONE]
𝒇 = a̅bc̅ + ab̅c̅ => c̅(a̅b + ab̅) => c̅(a⨁b) [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + abc [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + abc̅ => bc̅(a̅ + a) + ab̅c̅ => bc̅ + ab̅c̅ => c̅(b + ab̅) => c̅(b + a) => c̅b + c̅a [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + abc̅ + abc => bc̅(a̅ + a) + ab̅c̅ + abc => bc̅ + ab̅c̅ + abc => b(c̅ + ac) + ab̅c̅ => b(c̅ + a) + ab̅c̅ => bc̅ + ba + ab̅c̅ => ba + c̅(b + ab̅) => ba + c̅(b + a) => ba + c̅b + c̅a [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + ab̅c => a̅bc̅ + ab̅(c̅ + c) => a̅bc̅ + ab̅ [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + ab̅c + abc => a̅bc̅ + ab̅c̅ + ac(b̅ + b) => a̅bc̅ + ab̅c̅ + ac => a̅bc̅ + a(c + b̅c̅) => a̅bc̅ + a(c + b̅) => a̅bc̅ + ac + ab̅ [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + ab̅c + abc̅ => bc̅(a̅ + a) + ab̅(c̅ + c) => bc̅ + ab̅ [DONE]
𝒇 = a̅bc̅ + ab̅c̅ + ab̅c + abc̅ + abc => bc̅(a̅ + a) + ab̅c̅ + ab̅c + abc => bc̅ + ab̅c̅ + ab̅c + abc => c̅(b + ab̅) + ab̅c + abc => c̅(b + a) + ab̅c + abc => c̅b + c̅a + ab̅c + abc => c̅b + c̅a + ac(b̅ + b) => c̅b + c̅a + ac => c̅b + a(c + c̅) => c̅b + a [DONE]
𝒇 = a̅bc̅ + a̅bc => a̅b(c̅ + c) => a̅b [DONE]
𝒇 = a̅bc̅ + a̅bc + abc => a̅b(c̅ + c) + abc => a̅b + abc => b(a̅ + ac) => b(a̅ + c) => ba̅ + bc [DONE]
𝒇 = a̅bc̅ + a̅bc + abc̅ => a̅b(c̅ + c) + abc̅ => a̅b + abc̅ => b(a̅ + ac̅) => b(a̅ + c̅) => ba̅ + bc̅ [DONE]
𝒇 = a̅bc̅ + a̅bc + abc̅ + abc => a̅b(c̅ + c) + abc̅ + abc => a̅b + abc̅ + abc => b(a̅ + ac̅) + abc => b(a̅ + c̅) + abc => ba̅ + bc̅ + abc => b(a̅ + c̅ + ac) => b(a̅ + c̅ + c) => b(a̅ + 1) => b(1) => b [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c => a̅b(c̅ + c) + ab̅c => a̅b + ab̅c [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c + abc => a̅b(c̅ + c) + ab̅c + abc => a̅b + ab̅c + abc => a̅b + ac(b̅ + b) => a̅b + ac [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c + abc̅ => a̅b(c̅ + c) + ab̅c + abc̅ => a̅b + ab̅c + abc̅ => b(a̅ + ac̅) + ab̅c => b(a̅ + c̅) + ab̅c => ba̅ + bc̅ + ab̅c [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c + abc̅ + abc => a̅b(c̅ + c) + ab̅c + abc̅ + abc => a̅b + ab̅c + abc̅ + abc => a̅b + ab̅c + ab(c̅ + c) => a̅b + ab̅c + ab => a̅b + a(b̅c + b) => a̅b + a(b̅c + b) => a̅b + a[(b̅+b)(c+b)] => a̅b + ab + ac => b(a̅+a) + ac => b + ac [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ => a̅b(c̅ + c) + ab̅c̅ => a̅b + ab̅c̅ [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + abc => a̅b(c̅ + c) + ab̅c̅ + abc => a̅b + ab̅c̅ + abc => ab̅c̅ + b(ac + a̅) => ab̅c̅ + bc + ba̅ [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + abc̅ => a̅b(c̅ + c) + ab̅c̅ + abc̅ => a̅b + ab̅c̅ + abc̅ =>
a̅b + ab̅c̅ + abc̅ => a̅b + c̅(ab̅ + ab) => a̅b + c̅(ab̅ + ab) => a̅b + c̅[a(b̅ + b)] => a̅b + c̅a [CORRECT] [DONE]
a̅b + ab̅c̅ + abc̅ => ab̅c̅ + b(a̅ + ac̅) => ab̅c̅ + b[(a̅+a)(a̅c̅)] => ab̅c̅ + b[1(a̅c̅)] => ab̅c̅ + a̅bc̅ => c̅(ab̅ + a̅b) => c̅(a⨁b) [WRONG]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + abc̅ + abc => a̅bc̅ + a̅bc + ab̅c̅ + ab(c̅ + c) => a̅bc̅ + a̅bc + ab̅c̅ + abc̅ + ab => a̅bc̅ + a̅bc + ac̅(b̅ + b) + ab => a̅bc̅ + a̅bc + ac̅ + ab => a̅b(c̅ + c) + ac̅ + ab => a̅b + ac̅ + ab => ac̅ + b(a + a̅) => ac̅ + b [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + ab̅c => a̅b(c̅ + c) + ab̅(c̅ + c) => a̅b + ab̅ => a⨁b [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc => a̅b(c̅ + c) + ab̅(c̅ + c) + abc => a̅b + ab̅ + abc => a̅b + a(b̅ + bc) => a̅b + a[(b̅ + b)(b̅ + c)] => a̅b + a(b̅ + c) => a̅b + ab̅ + ac => a⨁b + ac [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅b(c̅ + c) + ab̅(c̅ + c) + abc̅ => a̅b + ab̅ + abc̅ => a̅b + a(b̅ + bc̅) => a̅b + a[(b̅ + b)(b̅ + c̅)] => a̅b + a(b̅ + c̅) => a̅b + ab̅ + ac̅ => a⨁b + ac̅ [DONE]
𝒇 = a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc => a̅b(c̅ + c) + ab̅(c̅ + c) + ab(c̅ + c) => a̅b + ab̅ + ab => a̅b + a(b̅ + b) => a̅b + a => (a + a̅)(a + b) => a + b [DONE]
𝒇 = a̅b̅c [DONE]
𝒇 = a̅b̅c + abc => c(a̅b̅ + ab) => c(a̅⨁̅b̅) => c(a⨀b) [DONE]
𝒇 = a̅b̅c + abc̅ [DONE]
𝒇 = a̅b̅c + abc̅ + abc => a̅b̅c + ab(c̅ + c) => a̅b̅c + ab [DONE]
𝒇 = a̅b̅c + ab̅c => b̅[a̅c + ac] => b̅c(a̅+a) => b̅c [DONE]
𝒇 = a̅b̅c + ab̅c + abc => b̅c(a̅ + a) + abc => b̅c + abc => c(b̅ + ab) => c[(b̅ + a)(b̅ + b)] => c(b̅ + a) => cb̅ + ca [DONE]
𝒇 = a̅b̅c + ab̅c + abc̅ => b̅c(a̅ + a) + abc̅ => b̅c + abc̅ [DONE]
𝒇 = a̅b̅c + ab̅c + abc̅ + abc => b̅(a̅c + ac) + ab(c̅ + c) => b̅[c(a̅ + a)] + ab => b̅c + ab [DONE]
𝒇 = a̅b̅c + ab̅c̅ => b̅(a̅c + ac̅) => b̅(a⨁c) [DONE]
𝒇 = a̅b̅c + ab̅c̅ + abc [DONE]
𝒇 = a̅b̅c + ab̅c̅ + abc̅ => a̅b̅c + ab̅c̅ + abc̅ => a̅b̅c + c̅(ab̅ + ab) => a̅b̅c + c̅[a(b̅ + b)] => a̅b̅c + c̅a [DONE]
𝒇 = a̅b̅c + ab̅c̅ + abc̅ + abc => a̅b̅c + ab̅c̅ + ab(c̅ + c) => a̅b̅c + ab̅c̅ + ab => a̅b̅c + a(b̅c̅ + b) => a̅b̅c + a[(b̅ + b)(b + c̅)] => a̅b̅c + a(b + c̅) => a̅b̅c + ab + ac̅ [DONE]
𝒇 = a̅b̅c + ab̅c̅ + ab̅c => b̅c(a̅ + a) + ab̅c̅ => b̅c + ab̅c̅ => b̅(c + ac̅) => b̅[(c + a)(c + c̅)] => b̅(c + a) => b̅c + b̅a [DONE]
𝒇 = a̅b̅c + ab̅c̅ + ab̅c + abc => a̅b̅c + ab̅(c̅ + c) + abc => a̅b̅c + ab̅ + abc => a̅b̅c + a(b̅ + bc) => a̅b̅c + a[(b̅ + b)(b̅ + c)] => a̅b̅c + a(b̅ + c) => a̅b̅c + ab̅ + ac => b̅(a̅c + a) + ac => b̅[(a̅ + a)(c + a)] + ac => b̅(c + a) + ac => b̅c + b̅a + ac [DONE]
𝒇 = a̅b̅c + ab̅c̅ + ab̅c + abc̅ => a̅b̅c + ab̅(c̅ + c) + abc̅ => a̅b̅c + ab̅ + abc̅ => b̅(a̅c + a) + abc̅ => b̅[(a̅ + a)(c + a)] + abc̅ => b̅(c + a) + abc̅ => b̅c + b̅a + abc̅ => b̅c + a(b̅ + bc̅) => b̅c + a[(b̅ + b)(b̅ + c̅)] => b̅c + a(b̅ + c̅) => b̅c + ab̅ + ac̅ {consensus rule} => b̅c + ac̅ [DONE]
𝒇 = a̅b̅c + ab̅c̅ + ab̅c + abc̅ + abc => a̅b̅c + ab̅(c̅ + c) + ab(c̅ + c) =>
a̅b̅c + ab̅ + ab => a̅b̅c + a(b̅ + b) => a̅b̅c + a [WRONG]
a̅b̅c + ab̅ + ab => b̅(a̅c + a) + ab => b̅[(a̅ + a)(c + a)] + ab => b̅(c + a) + ab => b̅c + b̅a + ab => b̅c + a(b̅ + b) => b̅c + a [CORRECT] [DONE]
𝒇 = a̅b̅c + a̅bc => a̅c(b̅ + b) => a̅c [DONE]
𝒇 = a̅b̅c + a̅bc + abc => a̅b̅c + bc(a̅ + a) => a̅b̅c + bc => c(b + a̅b̅) => c[(b + a̅)(b + b̅)] => c(b + a̅)
=> cb + ca̅ [DONE]
𝒇 = a̅b̅c + a̅bc + abc̅ => a̅c(b̅ + b) + abc̅ => a̅c + abc̅ [DONE]
𝒇 = a̅b̅c + a̅bc + abc̅ + abc => a̅c(b̅ + b) + ab(c̅ + c) => a̅c + ab [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c => a̅bc + b̅c(a̅ + a) => a̅bc + b̅c => c(a̅b + b̅) => c[(a̅ + b̅)(b + b̅)] => c(a̅ + b̅) => ca̅ + cb̅ [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c + abc => a̅c(b̅ + b) + ac(b̅ + b) => a̅c + ac => c(a̅ + a) => c [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c + abc̅ => a̅c(b̅ + b) + ab̅c + abc̅ => a̅c + ab̅c + abc̅ => c(a̅ + ab̅) + abc̅ => c[(a̅ + a)(a̅ + b̅)] + abc̅ => c(a̅ + b̅) + abc̅ => ca̅ + cb̅ + abc̅ [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c + abc̅ + abc => a̅c(b̅ + b) + ab̅c + ab(c̅ + c) => a̅c + ab̅c + ab =>
a̅c + ab̅c + ab => c(a̅ + ab̅) + ab => c[(a̅ + a)(a̅ + b̅)] + ab => c(a̅ + b̅) + ab => ca̅ + cb̅ + ab [WRONG]
a̅c + ab̅c + ab => a̅c + a(b̅c + b) => a̅c + a[(b̅ + b)(c + b)] => a̅c + a(c + b) => a̅c + ac + ab => c(a̅ + a) + ab => c + ab [CORRECT] [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ => a̅b̅c + a̅bc + ab̅c̅ => a̅c(b̅ + b) + ab̅c̅ => a̅c + ab̅c̅ [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + abc => a̅b̅c + a̅bc + ab̅c̅ + abc => a̅c(b̅ + b) + ab̅c̅ + abc => a̅c + ab̅c̅ + abc => ab̅c̅ + c(a̅ + ab) => ab̅c̅ + c[(a̅ + a)(a̅ + b)] => ab̅c̅ + c[(a̅ + a)(a̅ + b)] => ab̅c̅ + c(a̅ + b) => ab̅c̅ + ca̅ + cb [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + abc̅ => a̅b̅c + a̅bc + ab̅c̅ + abc̅ => a̅c(b̅ + b) + ac̅(b̅ + b) => a̅c + ac̅ => a⨁c [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + abc̅ + abc => a̅c(b̅ + b) + ac̅(b̅ + b) + abc => a̅c + ac̅ + abc =>
a̅c + ac̅ + abc => ac̅ + c(a̅ + ab) => ac̅ + c[(a̅ + a)(a̅ + b)] => ac̅ + c(a̅ + b) => ac̅ + ca̅ + cb
a̅c + ac̅ + abc => a̅c + a(c̅ + bc) => a̅c + a[(c̅ + b)(c̅ + c)] => a̅c + a(c̅ + b) => a̅c + ac̅ + ab [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + ab̅c => a̅c(b̅ + b) + ab̅(c̅ + c) => a̅c + ab̅ [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc => a̅c(b̅ + b) + ab̅(c̅ + c) + abc => a̅c + ab̅ + abc =>
a̅c + ab̅ + abc => c(a̅ + ab) + ab̅ => c[(a̅ + a)(a̅ + b)] + ab̅ => c(a̅ + b) + ab̅ => ca̅ + cb + ab̅ [WRONG]
a̅c + ab̅ + abc => a̅c + a(b̅ + bc) => a̅c + a[(b̅ + b)(b̅ + c)] => a̅c + a(b̅ + c) => a̅c + ab̅ + ac => ab̅ + c(a̅ + a) => ab̅ + c [CORRECT] [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅c(b̅ + b) + ab̅(c̅ + c) + abc̅ => a̅c + ab̅ + abc̅ => a̅c + a(b̅ + bc̅) => a̅c + a[(b̅ + b)(b̅ + c̅)] => a̅c + a(b̅ + c̅) => a̅c + ab̅ + ac̅ => ab̅ + a⨁c [DONE]
𝒇 = a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc => a̅c(b̅ + b) + ab̅(c̅ + c) + ab(c̅ + c) => a̅c + ab̅ + ab => a̅c + a(b̅ + b) => a̅c + a {absorption law} => c + a [DONE]
𝒇 = a̅b̅c + a̅bc̅ => a̅(b̅c + bc̅) => a̅(b⨁c) [DONE]
𝒇 = a̅b̅c + a̅bc̅ + abc =>
a̅b̅c + a̅bc̅ + abc => a̅(b̅c + bc̅) + abc => a̅(b⨁c) + abc
a̅b̅c + a̅bc̅ + abc => a̅b̅c + b(a̅c̅ + ac) => a̅b̅c + b(a⨀c) [DONE]
𝒇 = a̅b̅c + a̅bc̅ + abc̅ => a̅b̅c + bc̅(a̅ + a) => a̅b̅c + bc̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + abc̅ + abc => a̅b̅c + bc̅(a̅ + a) + abc => a̅b̅c + bc̅ + abc => a̅b̅c + b(c̅ + ac) => a̅b̅c + b[(c̅ + a)(c̅ + c)] => a̅b̅c + b(c̅ + a) => a̅b̅c + bc̅ + ba [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c => a̅b̅c + a̅bc̅ + ab̅c => a̅bc̅ + b̅c(a̅ + a) => a̅bc̅ + b̅c [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c + abc => a̅b̅c + a̅bc̅ + ac(b̅ + b) => a̅b̅c + a̅bc̅ + ac => c(a̅b̅ + a) + a̅bc̅ => c[(a̅ + a)(b̅ + a)] + a̅bc̅ => cb̅ + ca + a̅bc̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c + abc̅ => b̅c(a̅ + a) + bc̅(a̅ + a) => b̅c + bc̅ => b⨁c [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c + abc̅ + abc =>
b̅c(a̅ + a) + bc̅(a̅ + a) + abc => b̅c + bc̅ + abc => b̅c + b(c̅ + ac) => b̅c + b[(c̅ + a)(c̅ + c)] => b̅c + bc̅ + ba
b̅c(a̅ + a) + a̅bc̅ + ab(c̅ + c) => b̅c + a̅bc̅ + ab => b̅c + b(a̅c̅ + a) => b̅c + b[(a̅ + a)(c̅ + a)] => b̅c + bc̅ + ba [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ => a̅(b̅c + bc̅) + ab̅c̅ => a̅(b⨁c) + ab̅c̅ | a̅b̅c + c̅(a̅b + ab̅) => a̅b̅c + c̅(a⨁b) [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + abc => a̅(b̅c + bc̅) + a(b̅c̅ + bc) => a̅(b⨁c) + a(b̅⨁̅c̅) => a⨁b⨁c XOR [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + abc̅ => a̅b̅c + bc̅(a̅ + a) + ab̅c̅ => a̅b̅c + bc̅ + ab̅c̅ => a̅b̅c + c̅(b + ab̅) => a̅b̅c + c̅[(b + a)(b + b̅)] => a̅b̅c + c̅b + c̅a [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + abc̅ + abc =>
a̅b̅c + a̅bc̅ + ab̅c̅ + ab(c̅ + c) => a̅b̅c + a̅bc̅ + ab̅c̅ + ab => a̅b̅c + b(a̅c̅ + a) + ab̅c̅ => a̅b̅c + b[(a̅ + a)(c̅ + a)] + ab̅c̅ => a̅b̅c + b(c̅ + a) + ab̅c̅ => ab̅c̅ => a̅b̅c + bc̅ + ab + ab̅c̅
a̅b̅c + bc̅ + ab + ab̅c̅ => b̅(a̅c + c̅) + bc̅ + ba => b̅[(a̅ + c̅)(c + c̅)] + bc̅ + ba => b̅a̅ + b̅c̅ + bc̅ + ba => b̅a̅ + c̅(b̅ + b) + ba => b̅a̅ + c̅ + ab [DONE, SHORTEST ANSWER, WRONG]
a̅b̅c + bc̅ + ab + ab̅c̅ => a̅b̅c + c̅(b + ab̅) + ba => a̅b̅c + c̅(b + ab̅) + ba => a̅b̅c + c̅[(b + a)(b + b̅)] + ba => a̅b̅c + bc̅ + ac̅ + ab [DONE HERE, CORRECT]
a̅b̅c + bc̅(a̅ + a) + ab̅c̅ + abc => a̅b̅c + bc̅ + ab̅c̅ + abc =>
a̅b̅c + bc̅ + ab̅c̅ + abc => a̅b̅c + b(c̅ + ac) + ab̅c̅ => a̅b̅c + b[(c̅ + a)(c̅ + c)] + ab̅c̅ => a̅b̅c + bc̅ + ba + ab̅c̅ =>
a̅b̅c + bc̅ + ba + ab̅c̅ => a̅b̅c + c̅(b + ab̅) + ba => a̅b̅c + c̅[(b + a)(b + b̅)] + ba => a̅b̅c + bc̅ + ac̅ + ab [DONE HERE, CORRECT]
a̅b̅c + bc̅ + ba + ab̅c̅ => a̅b̅c + bc̅ + a(b + b̅c̅) => a̅b̅c + bc̅ + a[(b + b̅)(b + c̅)] => a̅b̅c + bc̅ + ac̅ + ab [DONE HERE, CORRECT]
a̅b̅c + bc̅ + ab̅c̅ + abc => a̅b̅c + c̅(b + ab̅) + abc => a̅b̅c + c̅[(b + a)(b + b̅)] + abc => a̅b̅c + bc̅ + ac̅ + abc =>
a̅b̅c + c̅b + c̅a + abc => a̅b̅c + c̅b + a(c̅ + bc) => a̅b̅c + c̅b + a[(c̅ + b)(c̅ + c)] => a̅b̅c + bc̅ + ac̅ + ab [TWO WAYS, SAME RES]
a̅b̅c + c̅b + c̅a + abc => a̅b̅c + b(c̅ + ac) + c̅a => a̅b̅c + b[(c̅ + a)(c̅ + c)] + c̅a => a̅b̅c + bc̅ + ac̅ + ab [DONE HERE, CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c => b̅c(a̅ + a) + a̅bc̅ + ab̅c̅ => b̅c + a̅bc̅ + ab̅c̅ => b̅(c + ac̅) + a̅bc̅ => b̅[(c + a)(c + c̅)] + a̅bc̅ => b̅c + b̅a + a̅bc̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc => b̅c(a̅ + a) + a̅bc̅ + ab̅c̅ + abc => b̅(c + ac̅) + a̅bc̅ + abc => b̅[(c + a)(c + c̅)] + a̅bc̅ + abc => b̅c + ab̅ + a̅bc̅ + abc => c(b̅ + ab) + ab̅ + a̅bc̅ => b̅c + ac + ab̅ + a̅bc̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ => b̅c(a̅ + a) + a̅bc̅ + ac̅(b̅ + b) => b̅c + a̅bc̅ + ac̅ => b̅c + c̅(a̅b + a) => b̅c + bc̅ + ac̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ + abc => b̅c(a̅ + a) + a̅bc̅ + ab̅c̅ + abc̅ + abc =>
|b̅c + a̅bc̅ + ab̅c̅ + abc̅ + abc => b̅c + a̅bc̅ + ab̅c̅ + ab(c̅ + c) => b̅c + a̅bc̅ + ab̅c̅ + ab =>
b̅c + a̅bc̅ + ab̅c̅ + ab => b̅(c + ac̅) + a̅bc̅ + ab => b̅c + ab̅ + a̅bc̅ + ab => b̅c + a(b̅ + b) + a̅bc̅ => b̅c + a + a̅bc̅ [DONE HERE]
b̅c + a̅bc̅ + ab̅c̅ + ab => b̅c + a̅bc̅ + a(b̅c̅ + b) => b̅c + ac̅ + a̅bc̅ + ab => b̅c + c̅(a + a̅b) + ab => b̅c + ac̅ + bc̅ + ab [DONE HERE]
|b̅c + a̅bc̅ + ab̅c̅ + abc̅ + abc => b̅c + a̅bc̅ + ac̅(b̅ + b) + abc => b̅c + a̅bc̅ + ac̅ + abc =>
b̅c + a̅bc̅ + ac̅ + abc => c(b̅ + ab) + a̅bc̅ + ac̅ => b̅c + ac + a̅bc̅ + ac̅ => b̅c + ac + c̅(a̅b + a) => b̅c + ac + bc̅ + ac̅ => b̅c + a(c + c̅) + bc̅ => b̅c + a + bc̅ => a + b⨁c [DONE HERE, CORRECT]
b̅c + a̅bc̅ + ac̅ + abc => b̅c + c̅(a + a̅b) + abc => b̅c + ac̅ + bc̅ + abc => b̅c + ac̅ + b(c̅ + ac) => b̅c + ac̅ + bc̅ + ab [DONE HERE, WRONG]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc => a̅b̅c + a̅b(c̅ + c) => a̅(b̅c + b) => a̅c + a̅b [DONE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + abc => a̅b̅c + a̅bc̅ + a̅bc + abc =>
a̅b̅c + a̅bc̅ + a̅bc + abc => a̅c(b̅ + b) + a̅bc̅ + abc => a̅c + a̅bc̅ + abc => c(a̅ + ab) + a̅bc̅ => a̅c + bc + a̅bc̅ => a̅(c + bc̅) + bc => a̅c + a̅b + bc [DONE, CORRECT]
a̅b̅c + a̅bc̅ + a̅bc + abc => a̅b̅c + a̅bc̅ + bc(a̅ + a) => a̅b̅c + a̅bc̅ + bc => c(a̅b̅ + b) + a̅bc̅ => a̅c + bc + a̅bc̅ => a̅c + b(c + a̅c̅) => a̅c + bc + a̅b [DONE, SAME ANSWER]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + abc̅ => a̅c(b̅ + b) + bc̅(a̅ + a) => a̅c + bc̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + abc̅ + abc =>
a̅b̅c + a̅bc̅ + a̅bc + abc̅ + abc => a̅c(b̅ + b) + bc̅(a̅ + a) + abc => a̅c + bc̅ + abc =>
a̅c + bc̅ + abc => c(a̅ + ab) + bc̅ => a̅c + bc + bc̅ => a̅c + b(c + c̅) => a̅c + b [DONE HERE, CORRECT]
a̅c + bc̅ + abc => a̅c + b(ac + c̅) => a̅c + ab + bc̅ [DONE HERE, WRONG]
a̅b̅c + a̅bc̅ + a̅bc + abc̅ + abc => a̅c(b̅ + b) + a̅bc̅ + ab(c̅ + c) => a̅c + a̅bc̅ + ab => a̅c + b(a̅c̅ + a) => a̅c + bc̅ + ab [DONE HERE, WRONG]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c => b̅c(a + a̅) + a̅b(c̅ + c) => b̅c + a̅b [DONE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc =>
b̅c(a̅ + a) + a̅b(c̅ + c) + abc => b̅c + a̅b + abc =>
c(b̅ + ab) + a̅b => b̅c + ac + a̅b [DONE HERE, WRONG]
b̅c + b(a̅ + ac) => b̅c + a̅b + bc [DONE HERE, WRONG]
a̅c(b̅ + b) + a̅bc̅ + ac(b̅ + b) => a̅c + a̅bc̅ + ac =>
c(a̅ + a) + a̅bc̅ => c + a̅bc̅ => (c + a̅b)(c + c̅) => c + a̅b [DONE HERE, CORRECT]
a̅(c + bc̅) + ac => a̅c + a̅b + ac => c(a̅ + a) + a̅b => c + a̅b [DONE HERE, CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc̅ => a̅c(b̅ + b) + bc̅(a̅ + a) + ab̅c => a̅c + bc̅ + ab̅c => c(a̅ + ab̅) + bc̅ => a̅c + b̅c + bc̅ == a̅c + b⨁c [DONE]
a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc̅ => a̅b(c̅ + c) + b̅c(a̅ + a) + abc̅ => b(a̅ + ac̅) + b̅c => a̅b + bc̅ + b̅c == a̅b + b⨁c [DONE, ALSO CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc̅ + abc =>
a̅b̅c + a̅b(c̅ + c) + ab̅c + ab(c̅ + c) => a̅b̅c + a̅b + ab̅c + ab => a̅(b̅c + b) + a(b̅c + b) => a̅(c + b) + a(c + b) => a̅c + a̅b + ac + ab => c(a̅ + a) + b(a̅ + a) => c + b [DONE HERE, CORRECT]
a̅c(b̅ + b) + bc̅(a̅ + a) + ac(b̅ + b) => a̅c(b̅ + b) + bc̅(a̅ + a) + ac(b̅ + b) => c(a̅ + a) + bc̅ => c + bc̅ [DONE HERE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ => a̅c(b̅ + b) + a̅bc̅ + ab̅c̅ => a̅(c + bc̅) + ab̅c̅ => a̅c + a̅b + ab̅c̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc => a̅b̅c + a̅bc̅ + bc(a̅ + a) + ab̅c̅ => a̅b̅c + a̅bc̅ + bc + ab̅c̅ => a̅b̅c + b(a̅c̅ + c) + ab̅c̅ => a̅b̅c + a̅b + bc + ab̅c̅ => a̅(b̅c + b) + bc + ab̅c̅ => a̅c + a̅b + bc + ab̅c̅ [DONE]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ => a̅c(b̅ + b) + a̅bc̅ + ac̅(b̅ + b) => a̅c + a̅bc̅ + ac̅ =>
a̅(c + bc̅) + ac̅ => a̅c + a̅b + ac̅ == a̅b + a⨁c [DONE, CORRECT]
a̅c + c̅(a̅b + a) => a̅c + bc̅ + ac̅ == bc̅ + a⨁c [DONE, ALSO CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ + abc => a̅c(b̅ + b) + a̅bc̅ + ab̅c̅ + abc̅ + abc =>
a̅c(b̅ + b) + bc̅(a̅ + a) + ab̅c̅ + abc => a̅c + c̅(b + ab̅) + abc => a̅c + bc̅ + ac̅ + abc =>
c(a̅ + ab) + bc̅ + ac̅ => a̅c + bc + bc̅ + ac̅ => a̅c + b(c + c̅) + ac̅ => a̅c + b + ac̅ == a⨁c + b [DONE HERE, CORRECT]
a̅c + b(c̅ + ac) + ac̅ => a̅c + bc̅ + ab + ac̅ => [DONE HERE, WRONG]
a̅c(b̅ + b) + a̅bc̅ + ab̅c̅ + ab(c̅ + c) => a̅(c + bc̅) + a(b̅c̅ + b) => a̅c + a̅b + ac̅ + ab => a̅c + b(a̅ + a) + ac̅ => a̅c + b + ac̅ == a⨁c + b [DONE HERE, CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c => a̅b̅c + a̅b(c̅ + c) + ab̅(c̅ + c) => a̅b̅c + a̅b + ab̅ =>
a̅(b̅c + b) + ab̅ => a̅c + a̅b + ab̅ == a̅c + a⨁b [DONE HERE, CORRECT]
b̅(a̅c + a) + a̅b => b̅c + ab̅ + a̅b == b̅c + a⨁b [DONE HERE, ALSO CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc =>
a̅b̅c + a̅b(c̅ + c) + ab̅c̅ + ab̅(c + c) => a̅b̅c + a̅b + ab̅c̅ + ab̅ =>
a̅(b̅c + b) + ab̅c̅ + ab̅ => a̅c + a̅b + ab̅c̅ + ab̅ => [DONE HERE]
b̅(a̅c + a) + a̅b + ab̅c̅ => b̅c + ab̅ + a̅b + ab̅c̅ => b̅c + ab̅ + a̅b + ab̅c̅ => b̅(c + ac̅) + ab̅ + a̅b => b̅(c + a) + ab̅ + a̅b => b̅c + ab̅ + ab̅ + a̅b => b̅c + ab̅ + a̅b [DONE HERE]
|
a̅b̅c + a̅b(c̅ + c) + ab̅(c̅ + c) + abc => a̅b̅c + a̅b + ab̅ + abc =>
a̅(b̅c + b) + a(b̅ + bc) => a̅(c + b) + a(b̅ + c) => a̅c + a̅b + ab̅ + ac => c(a̅ + a) + a̅b + ab̅ => c + a̅b + ab̅ == c + a⨁b [DONE HERE, CORRECT]
b̅(a̅c + a) + b(a̅ + ac) => b̅(c + a) + b(a̅ + c) => b̅c + ab̅ + a̅b + bc => c(b̅ + b) + ab̅ + a̅b => c + ab̅ + a̅b == c + a⨁b [DONE HERE, SAME ANSWER]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ =>
a̅b̅c + a̅b(c̅ + c) + ab̅(c̅ + c) + abc̅ => a̅(b̅c + b) + a(b̅ + bc̅) => a̅(c + b) + a(b̅ + c̅) => a̅c + a̅b + ab̅ + ac̅ => a̅c + a̅b + ab̅ + ac̅ [DONE HERE, WRONG]
a̅c(b̅ + b) + bc̅(a̅ + a) + ab̅(c̅ + c) => a̅c + bc̅ + ab̅ [DONE HERE, CORRECT]
b̅c(a̅ + a) + a̅b(c̅ + c) + ac̅(b̅ + b) => b̅c + a̅b + ac̅ [DONE HERE, ALSO CORRECT]
𝒇 = a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc => a̅b̅c + a̅b(c̅ + c) + ab̅(c̅ + c) + ab(c̅ + c) => a̅b̅c + a̅b + ab̅ + ab =>
a̅b̅c + a̅b + ab̅ + ab => a̅(b̅c + b) + a(b̅ + b) => a̅c + a̅b + a => a̅c + (a̅ + a)(b + a) => a̅c + b + a => (a̅ + a)(c + a) + b => c + a + b [DONE HERE, CORRECT]
a̅b̅c + a̅b + ab̅ + ab => b̅(a̅c + a) + b(a̅ + a) => b̅c + ab̅ + b => c + a + b [DONE HERE, SAME ANSWER]
𝒇 = a̅b̅c̅ [DONE]
𝒇 = a̅b̅c̅ + abc [DONE]
𝒇 = a̅b̅c̅ + abc̅ => c̅(a̅b̅ + ab) => c̅(a̅⨁̅b̅) => c̅(a⨀b) [DONE]
𝒇 = a̅b̅c̅ + abc̅ + abc => a̅b̅c̅ + ab(c̅ + c) => a̅b̅c̅ + ab [DONE]
𝒇 = a̅b̅c̅ + ab̅c => b̅(a̅c̅ + ac) => b̅(a̅⨁̅c̅) => b̅(a⨀c) [DONE]
𝒇 = a̅b̅c̅ + ab̅c + abc => a̅b̅c̅ + ac(b̅ + b) => a̅b̅c̅ + ac [DONE]
𝒇 = a̅b̅c̅ + ab̅c + abc̅ [DONE]
𝒇 = a̅b̅c̅ + ab̅c + abc̅ + abc => a̅b̅c̅ + ab̅c + abc̅ + abc =>
a̅b̅c̅ + ac(b̅ + b) + abc̅ => a̅b̅c̅ + ac + abc̅ => a̅b̅c̅ + a(c + bc̅) => a̅b̅c̅ + a(c + b) => a̅b̅c̅ + ac + ab [DONE]
a̅b̅c̅ + ab̅c + ab(c̅ + c) => a̅b̅c̅ + ab̅c + ab => a̅b̅c̅ + a(b̅c + b) => a̅b̅c̅ + a(c + b) => a̅b̅c̅ + ac + ab [DONE]
𝒇 = a̅b̅c̅ + ab̅c̅ => b̅c̅(a̅ + a) => b̅c̅ [DONE]
𝒇 = a̅b̅c̅ + ab̅c̅ + abc => b̅c̅(a̅ + a) + abc => b̅c̅ + abc [DONE]
𝒇 = a̅b̅c̅ + ab̅c̅ + abc̅ =>
a̅b̅c̅ + ab̅c̅ + abc̅ => b̅c̅(a̅ + a) + abc̅ => b̅c̅ + abc̅ => c̅(b̅ + ab) => c̅[(b̅ + a)(b̅ + b)] => b̅c̅ + ac̅ [DONE]
a̅b̅c̅ + ab̅c̅ + abc̅ => a̅b̅c̅ + ac̅(b̅ + b) => a̅b̅c̅ + ac̅ => c̅(a̅b̅ + a) => c̅[(a̅ + a)(b̅ + a)] => b̅c̅ + ac̅ [SAME ANSWER]
𝒇 = a̅b̅c̅ + ab̅c̅ + abc̅ + abc => b̅c̅(a̅ + a) + ab(c̅ + c) => b̅c̅ + ab [DONE]
𝒇 = a̅b̅c̅ + ab̅c̅ + ab̅c =>
a̅b̅c̅ + ab̅c̅ + ab̅c => b̅c̅(a̅ + a) + ab̅c => b̅c̅ + ab̅c => b̅(c̅ + ac) => b̅[(c̅ + a)(c̅ + c)] => b̅c̅ + ab̅ => [DONE]
a̅b̅c̅ + ab̅c̅ + ab̅c => a̅b̅c̅ + ab̅(c̅ + c) => a̅b̅c̅ + ab̅ => b̅(a̅c̅ + a) => b̅[(a̅ + a)(c̅ + a)] => b̅c̅ + ab̅ => [SAME ANSWER]
𝒇 = a̅b̅c̅ + ab̅c̅ + ab̅c + abc => b̅c̅(a̅ + a) + ac(b̅ + b) => b̅c̅ + ac [DONE]
𝒇 = a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ =>
a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ => b̅c̅(a̅ + a) + ab̅c + abc̅ => b̅c̅ + ab̅c + abc̅ =>
b̅c̅ + ab̅c + abc̅ => b̅(c̅ + ac) + abc̅ => b̅[(c̅ + a)(c̅ + c)] + abc̅ => b̅c̅ + ab̅ + abc̅ => b̅c̅ + a(b̅ + bc̅) => b̅c̅ + ab̅ + ac̅ [DONE]
b̅c̅ + ab̅c + abc̅ => c̅(b̅ + ab) + ab̅c => c̅[(b̅ + a)(b̅ + b)] + ab̅c => b̅c̅ + ac̅ + ab̅c => b̅c̅ + a(c̅ + b̅c) => b̅c̅ + ab̅ + ac̅ [SAME ANSWER]
a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ => a̅b̅c̅ + ab̅(c̅ + c) + abc̅ => a̅b̅c̅ + ab̅ + abc̅ =>
a̅b̅c̅ + ab̅ + abc̅ => b̅(a̅c̅ + a) + abc̅ => b̅(a̅c̅ + a) + abc̅ => b̅c̅ + b̅a + abc̅ => c̅(b̅ + ab) + b̅a =>
a̅b̅c̅ + ab̅ + abc̅ => a̅b̅c̅ + a(b̅ + bc̅) => a̅b̅c̅ + a(b̅ + bc̅) => a̅b̅c̅ + ab̅ + ac̅ =>
a̅b̅c̅ + ab̅ + ac̅ => b̅(a̅c̅ + a) + ac̅ => b̅c̅ + ab̅ + ac̅ => [SAME ANSWER]
a̅b̅c̅ + ab̅ + ac̅ => c̅(a̅b̅ + a) + ab̅ => b̅c̅ + ab̅ + ac̅ => [SAME ANSWER]
𝒇 = a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ + abc =>
a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ + abc => b̅c̅(a̅ + a) + ab̅c + ab(c̅ + c) => b̅c̅ + ab̅c + ab =>
b̅c̅ + ab̅c + ab => b̅(c̅ + ac) + ab => b̅c̅ + ab̅ + ab => b̅c̅ + a(b̅ + b) => b̅c̅ + a [DONE, CORRECT]
b̅c̅ + ab̅c + ab => b̅c̅ + a(b̅c + b) => b̅c̅ + ac + ab [DONE, WRONG]
a̅b̅c̅ + ab̅c̅ + ab̅c + abc̅ + abc => a̅b̅c̅ + ab̅(c̅ + c) + ab(c̅ + c) => a̅b̅c̅ + ab̅ + ab =>
a̅b̅c̅ + ab̅ + ab => a̅b̅c̅ + a(b̅ + b) => a̅b̅c̅ + a [DONE, WRONG]
a̅b̅c̅ + ab̅ + ab => b̅(a̅c̅ + a) + ab => b̅c̅ + ab̅ + ab => b̅c̅ + a(b̅ + b) => b̅c̅ + a [DONE, CORRECT]
𝒇 = a̅b̅c̅ + a̅bc => a̅(b̅c̅ + bc) => a̅(b̅⨁̅c̅) => a̅(b⨀c) [DONE]
𝒇 = a̅b̅c̅ + a̅bc + abc => a̅b̅c̅ + bc(a̅ + a) => a̅b̅c̅ + bc [DONE]
𝒇 = a̅b̅c̅ + a̅bc + abc̅ =>
a̅b̅c̅ + a̅bc + abc̅ => a̅b̅c̅ + b(a̅c + ac̅) => a̅b̅c̅ + b(a⨁c) [DONE]
a̅b̅c̅ + a̅bc + abc̅ => c̅(a̅b̅ + ab) + a̅bc => c̅(a̅⨁̅b̅) + a̅bc => c̅(a⨀b) + a̅bc [DONE]
𝒇 = a̅b̅c̅ + a̅bc + abc̅ + abc => a̅b̅c̅ + abc̅ + bc(a̅+a) => a̅b̅c̅ + abc̅ + bc => a̅b̅c̅ + b(ac̅ + c) => a̅b̅c̅ + b[(a + c)(c̅ + c)] => a̅b̅c̅ + ab + bc [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c => a̅(b̅c̅ + bc) + ab̅c => a̅(b̅⨁̅c̅) + ab̅c => a̅(b⨀c) + ab̅c [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c + abc => a̅b̅c̅ + a̅bc + ac(b̅ + b) => a̅b̅c̅ + a̅bc + ac => a̅b̅c̅ + c(a̅b + a) => a̅b̅c̅ + bc + ac [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c + abc̅ => a̅(b̅c̅ + bc) + a(b̅c + bc̅) => a̅(b̅⨁̅c̅) + a(b⨁c) => a̅(b⨀c) + a(b⨁c) => a̅⨁̅b̅⨁̅c̅, a⨀b⨀c XNOR [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c + abc̅ + abc => a̅b̅c̅ + a̅bc + ab̅c + ab(c̅ + c) => a̅b̅c̅ + ab̅c + b(a̅c + a) => a̅b̅c̅ + ab̅c + bc + ab => a̅b̅c̅ + bc + a(b̅c + b) => a̅b̅c̅ + bc + ac + ab [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ => b̅c̅(a̅ + a) + a̅bc => b̅c̅ + a̅bc [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + abc => b̅c̅(a̅ + a) + bc(a̅ + a) => b̅c̅ + bc => b̅⨁̅c̅, b⨀c [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + abc̅ =>
b̅c̅(a̅ + a) + a̅bc + abc̅ => b̅c̅ + a̅bc + abc̅ => a̅bc + c̅(b̅ + ab) => a̅bc + b̅c̅ + ac̅ => [DONE]
a̅b̅c̅ + a̅bc + ac̅(b̅ + b) => a̅b̅c̅ + a̅bc + ac̅ => a̅bc + c̅(a + a̅b̅) => a̅bc + ac̅ + b̅c̅ => [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + abc̅ + abc =>
b̅c̅(a̅ + a) + a̅bc + ab(c̅ + c) => b̅c̅ + a̅bc + ab => b̅c̅ + b(a̅c + a) => b̅c̅ + bc + ab => [DONE]
a̅b̅c̅ + bc(a̅ + a) + ac̅(b̅ + b) => a̅b̅c̅ + bc + ac̅ => bc + c̅(a + a̅b̅) => bc + ac̅ + b̅c̅ => [DONE, ALSO CORRECT]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + ab̅c => a̅b̅c̅ + a̅bc + ab̅(c̅ + c) => a̅b̅c̅ + a̅bc + ab̅ => a̅bc + b̅(a + a̅c̅) => a̅bc + ab̅ + b̅c̅ [DONE]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + ab̅c + abc =>
a̅b̅c̅ + ab̅(c̅ + c) + bc(a̅ + a) => a̅b̅c̅ + ab̅ + bc => b̅(a + a̅c̅) + bc => ab̅ + b̅c̅ + bc [DONE]
a̅bc + b̅c̅(a̅ + a) + ac(b̅ + b) => a̅bc + b̅c̅ + ac => b̅c̅ + c(a + a̅b) => b̅c̅ + ac + bc [DONE, ALSO CORRECT]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ =>
b̅c̅(a̅ + a) + a̅bc + ab̅c + abc̅ => b̅c̅ + a̅bc + ab̅c + abc̅ =>
b̅(c̅ + ac) + a̅bc + abc̅ => b̅c̅ + ab̅ + a̅bc + abc̅ => b̅c̅ + a(b̅ + bc̅) + a̅bc => b̅c̅ + ab̅ + ac̅ + a̅bc [DONE]
c̅(b̅ + ab) + a̅bc + ab̅c => b̅c̅ + ac̅ + a̅bc + ab̅c => b̅c̅ + a(c̅ + b̅c) + a̅bc => b̅c̅ + ac̅ + ab̅ + a̅bc [DONE, SAME ANSWER]
a̅b̅c̅ + a̅bc + ab̅(c̅ + c) + abc̅ => a̅b̅c̅ + a̅bc + ab̅ + abc̅ =>
a̅b̅c̅ + a̅bc + a(b̅ + bc̅) => a̅b̅c̅ + a̅bc + ab̅ + ac̅ => a̅bc + ab̅ + c̅(a + a̅b̅) => a̅bc + ab̅ + ac̅ + b̅c̅ [DONE, SAME ANSWER]
a̅bc + b̅(a + a̅c̅) + abc̅ => a̅bc + ab̅ + b̅c̅ + abc̅ => a̅bc + a(b̅ + bc̅) + b̅c̅ => a̅bc + ab̅ + ac̅ + b̅c̅ [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc =>
a̅b̅c̅ + bc(a̅ + a) + ac̅(b̅ + b) + ab̅c => a̅b̅c̅ + bc + ac̅ + ab̅c => a̅b̅c̅ + bc + ac̅ + ab̅c =>
bc + c̅(a + a̅b̅) + ab̅c => bc + ac̅ + b̅c̅ + ab̅c => bc + ac̅ + b̅c̅ + ab̅c =>
ac̅ + b̅c̅ + c(b + ab̅) => ac̅ + b̅c̅ + bc + ac [DONE, WRONG]
bc + a(c̅ + b̅c) + b̅c̅ => bc + ac̅ + ab̅ + b̅c̅ [DONE, WRONG]
a̅b̅c̅ + bc + a(c̅ + b̅c) => a̅b̅c̅ + bc + ac̅ + ab̅ => a̅b̅c̅ + bc + ac̅ + ab̅ =>
bc + c̅(a + a̅b̅) + ab̅ => bc + ac̅ + b̅c̅ + ab̅ [DONE, WRONG]
bc + ac̅ + b̅(a + a̅c̅) => bc + ac̅ + ab̅ + b̅c̅ [DONE, WRONG]
b̅c̅(a̅ + a) + a̅bc + ab̅c + ab(c̅ + c) => b̅c̅ + a̅bc + ab̅c + ab =>
b̅(c̅ + ac) + b(a + a̅c) => b̅c̅ + ab̅ + ab + bc => b̅c̅ + a(b̅ + b) + bc => b̅c̅ + a + bc => a + b̅⨁̅c̅ => a + b⨀c [DONE, CORRECT]
b̅c̅ + a̅bc + a(b̅c + b) => b̅c̅ + a̅bc + a(c + b) => b̅c̅ + a̅bc + ac + ab =>
b̅c̅ + c(a̅b + a) + ab => b̅c̅ + bc + ac + ab [DONE, WRONG]
b̅c̅ + ac + b(a + a̅c) => b̅c̅ + ac + ab + bc [DONE, WRONG]
a̅b̅c̅ + a̅bc + ab̅(c̅ + c) + ab(c̅ + c) => b̅(a + a̅c̅) + b(a + a̅c) => ab̅ + b̅c̅ + ab + bc => a(b̅ + b) + b̅c̅ + bc => a + b̅c̅ + bc => a + b̅c̅ + bc => a + b̅⨁̅c̅ => a + b⨀c [DONE, CORRECT]
𝒇 = a̅b̅c̅ + a̅bc̅ => a̅c̅(b̅ + b) => a̅c̅ [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + abc => a̅c̅(b̅ + b) + abc => a̅c̅ + abc [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + abc̅ => a̅b̅c̅ + a̅bc̅ + abc̅ =>
a̅c̅(b̅ + b) + abc̅ => a̅c̅ + abc̅ => c̅(a̅ + ab) => a̅c̅ + bc̅ [DONE]
a̅b̅c̅ + bc̅(a̅ + a) => a̅b̅c̅ + bc̅ => c̅(a̅b̅ + b) => a̅c̅ + bc̅ [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc̅ + abc̅ + abc =>
a̅c̅(b̅ + b) + ab(c̅ + c) => a̅c̅ + ab [DONE]
a̅b̅c̅ + bc̅(a̅ + a) + abc => a̅b̅c̅ + bc̅ + abc =>
a̅b̅c̅ + bc̅ + abc => c̅(a̅b̅ + b) + abc => a̅c̅ + bc̅ + abc => a̅c̅ + b(c̅ + ac) => a̅c̅ + b(c̅ + a) => a̅c̅ + bc̅ + ab [DONE, WRONG]
a̅b̅c̅ + bc̅ + abc => a̅b̅c̅ + b(c̅ + ac) => a̅b̅c̅ + bc̅ + ab => c̅(a̅b̅ + b) + ab => c̅(a̅ + b) + ab => a̅c̅ + bc̅ + ab [DONE, WRONG]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c => a̅c̅(b̅ + b) + ab̅c => a̅c̅ + ab̅c [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c + abc => a̅b̅c̅ + a̅bc̅ + ab̅c + abc => a̅c̅(b̅ + b) + ac(b̅ + b) => a̅c̅ + ac => a̅⨁̅c̅ => a⨀c XNOR [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c + abc̅ => a̅c̅(b̅ + b) + ab̅c + abc̅ => a̅c̅ + ab̅c + abc̅ => c̅(a̅ + ab) + ab̅c => a̅c̅ + bc̅ + ab̅c [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c + abc̅ + abc =>
a̅b̅c̅ + a̅bc̅ + ab̅c + abc̅ + abc => a̅c̅(b̅ + b) + ab̅c + ab(c̅ + c) => a̅c̅ + ab̅c + ab => a̅c̅ + a(b̅c + b) => a̅c̅ + a(c + b) => a̅c̅ + ac + ab [DONE, CORRECT]
a̅b̅c̅ + a̅bc̅ + ab̅c + abc̅ + abc => a̅b̅c̅ + ac(b̅ + b) + bc̅(a̅ + a) => a̅b̅c̅ + ac + bc̅ => ac + c̅(b + a̅b̅) => ac + c̅(b + a̅) => ac + bc̅ + a̅c̅ => [DONE, ALSO CORRECT]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ =>
a̅b̅c̅ + a̅bc̅ + ab̅c̅ => a̅c̅(b̅ + b) + ab̅c̅ => a̅c̅ + ab̅c̅ => c̅(a̅ + b̅) => a̅c̅ + b̅c̅ [DONE]
a̅b̅c̅ + a̅bc̅ + ab̅c̅ => b̅c̅(a̅ + a) + a̅bc̅ => b̅c̅ + a̅bc̅ => c̅(b̅ + a̅) => a̅c̅ + b̅c̅ [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + abc => b̅c̅(a̅ + a) + a̅bc̅ + abc => b̅c̅ + a̅bc̅ + abc => c̅(b̅ + a̅b) + abc => b̅c̅ + a̅c̅ + abc [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + abc̅ => a̅c̅(b̅ + b) + ac̅(b̅ + b) => a̅c̅ + ac̅ => c̅(a̅ + a) => c̅ [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + abc̅ + abc => a̅c̅(b̅ + b) + ac̅(b̅ + b) + abc => a̅c̅ + ac̅ + abc => a̅c̅ + ac̅ + ab => c̅(a̅ + a) + ab => c̅ + ab [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c => a̅c̅(b̅ + b) + ab̅(c̅ + c) => a̅c̅ + ab̅ [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc =>
a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc => a̅c̅(b̅ + b) + ab̅(c̅ + c) + abc => a̅c̅ + ab̅ + abc => a̅c̅ + a(b̅ + bc) => a̅c̅ + ab̅ + ac [DONE, CORRECT]
a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc => b̅c̅(a̅ + a) + a̅bc̅ + ac(b̅ + b) => b̅c̅ + a̅bc̅ + ac => c̅(b̅ + a̅b) + ac => b̅c̅ + a̅c̅ + ac [DONE, ALSO CORRECT]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ =>
a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ => a̅c̅(b̅ + b) + ab̅(c̅ + c) + abc̅ => a̅c̅ + ab̅ + abc̅ =>
a̅c̅ + ab̅ + abc̅ => c̅(a̅ + ab) + ab̅ => c̅(a̅ + b) + ab̅ => a̅c̅ + bc̅ + ab̅ [DONE, WRONG]
a̅c̅ + ab̅ + abc̅ => a̅c̅ + a(b̅ + bc̅) => a̅c̅ + a(b̅ + c̅) => a̅c̅ + ab̅ + ac̅ => c̅(a̅ + a) + ab̅ => c̅ + ab̅ [DONE, CORRECT]
a̅b̅c̅ + ab̅(c̅ + c) + bc̅(a̅ + a) => a̅b̅c̅ + ab̅ + bc̅ =>
a̅b̅c̅ + ab̅ + bc̅ => b̅(a̅c̅ + a) + bc̅ => b̅(c̅ + a) + bc̅ => b̅c̅ + ab̅ + bc̅ => c̅(b̅ + b) + ab̅ => c̅ + ab̅ [DONE, CORRECT]
a̅b̅c̅ + ab̅ + bc̅ => c̅(a̅b̅ + b) + ab̅ => c̅(a̅ + b) + ab̅ => a̅c̅ + bc̅ + ab̅ [DONE, WRONG]
𝒇 = a̅b̅c̅ + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ + abc => a̅c̅(b̅ + b) + ab̅(c̅ + c) + ab(c̅ + c) => a̅c̅ + ab̅ + ab => a̅c̅ + a(b̅ + b) => a̅c̅ + a => c̅ + a [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc =>
a̅c̅(b̅ + b) + a̅bc => a̅c̅ + a̅bc => a̅(c̅ + bc) => a̅c̅ + a̅b [DONE]
a̅b̅c̅ + a̅b(c̅ + c) => a̅b̅c̅ + a̅b => a̅(b̅c̅ + b) => a̅c̅ + a̅b [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + abc => a̅c̅(b̅ + b) + bc(a̅ + a) => a̅c̅ + bc [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + abc̅ =>
a̅b̅c̅ + a̅bc̅ + a̅bc + abc̅ => a̅c̅(b̅ + b) + a̅bc + abc̅ => a̅c̅ + a̅bc + abc̅ =>
a̅c̅ + a̅bc + abc̅ => a̅(c̅ + bc) + abc̅ => a̅(c̅ + b) + abc̅ => a̅c̅ + a̅b + abc̅ => a̅c̅ + b(a̅ + ac̅) => a̅c̅ + a̅b + bc̅ [DONE]
a̅c̅ + a̅bc + abc̅ => c̅(a̅ + ab) + a̅bc => c̅(a̅ + b) + a̅bc => a̅c̅ + bc̅ + a̅bc => a̅(c̅ + bc) + bc̅ => a̅c̅ + a̅b + bc̅ [DONE, SAME ANSWER]
a̅b̅c̅ + a̅bc̅ + a̅bc + abc̅ => a̅b̅c̅ + a̅b(c̅ + c) + abc̅ => a̅b̅c̅ + a̅b + abc̅ =>
a̅b̅c̅ + a̅b + abc̅ => a̅(b̅c̅ + b) + abc̅ => a̅c̅ + a̅b + abc̅ => a̅c̅ + a̅b + abc̅ =>
a̅c̅ + a̅b + abc̅ => a̅c̅ + b(a̅ + ac̅) => a̅c̅ + b(a̅ + c̅) => a̅c̅ + a̅b + bc̅ [DONE, SAME ANSWER]
a̅c̅ + a̅b + abc̅ => c̅(a̅ + ab) + a̅b => c̅(a̅ + b) + a̅b => a̅c̅ + bc̅ + a̅b [DONE, SAME ANSWER]
a̅b̅c̅ + a̅b + abc̅ => a̅b̅c̅ + b(a̅ + ac̅) => a̅b̅c̅ + a̅b + bc̅ => a̅b̅c̅ + a̅b + bc̅ =>
a̅b̅c̅ + a̅b + bc̅ => a̅(b̅c̅ + b) + bc̅ => a̅(c̅ + b) + bc̅ => a̅c̅ + a̅b + bc̅ [DONE, SAME ANSWER]
a̅b̅c̅ + a̅b + bc̅ => c̅(a̅b̅ + b) + a̅b => c̅(a̅ + b) + a̅b => a̅c̅ + bc̅ + a̅b [DONE, SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + abc̅ + abc =>
a̅c̅(b̅ + b) + a̅bc + ab(c̅ + c) => a̅c̅ + a̅bc + ab =>
a̅c̅ + a̅bc + ab => a̅(c̅ + bc) + ab => a̅c̅ + a̅b + ab => a̅c̅ + b(a̅ + a) => a̅c̅ + b [DONE]
a̅c̅ + a̅bc + ab => a̅c̅ + b(a̅c + a) => a̅c̅ + bc + ab [DONE, WRONG]
a̅b̅c̅ + a̅b(c̅ + c) + ab(c̅ + c) => a̅b̅c̅ + a̅b + ab => a̅(c̅ + b) + ab => a̅c̅ + a̅b + ab => a̅c̅ + b(a̅ + a) => a̅c̅ + b [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c => a̅b̅c̅ + a̅b(c̅ + c) + ab̅c => a̅(b̅c̅ + b) + ab̅c => a̅c̅ + a̅b + ab̅c [DONE]
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + abc =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c => a̅b̅(c̅ + c) => a̅b̅ => ā+̄b̄ NOR [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + abc => a̅b̅(c̅ + c) + abc => a̅b̅ + abc [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + abc̅ => a̅b̅(c̅ + c) + abc̅ => a̅b̅ + abc̅ [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + abc̅ + abc => a̅b̅(c̅ + c) + ab(c̅ + c) => a̅b̅ + ab => a̅⨁̅b̅ => a⨀b XNOR [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c =>
a̅b̅c̅ + a̅b̅c + ab̅c => a̅b̅(c̅ + c) + ab̅c => a̅b̅ + ab̅c => b̅(a̅ + ac) => a̅b̅ + b̅c [DONE]
a̅b̅c̅ + a̅b̅c + ab̅c => a̅b̅c̅ + b̅c(a̅ + a) => a̅b̅c̅ + b̅c => b̅(a̅c̅ + c) => a̅b̅ + b̅c [SAME ANSWER]
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ => a̅b̅(c̅ + c) + ab̅c̅ => a̅b̅ + ab̅c̅ => b̅(a̅ + ac̅) => a̅b̅ + b̅c̅ [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + ab̅c̅ + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc => a̅b̅(c̅ + c) + a̅bc => a̅b̅ + a̅bc => a̅(b̅ + bc) => a̅b̅ + a̅c [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc + ab̅c̅ + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ => a̅b̅(c̅ + c) + a̅bc̅ => a̅b̅ + a̅bc̅ => a̅(b̅ + bc̅) => a̅b̅ + a̅c̅ [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + ab̅c̅ + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc => a̅b̅(c̅ + c) + a̅b(c̅ + c) => a̅b̅ + a̅b => a̅(b̅ + b) => a̅ [DONE]
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + abc̅ + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc =>
𝒇 = a̅b̅c̅ + a̅b̅c + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅b̅(c̅ + c) + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅b̅ + a̅bc̅ + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅(b̅ + bc̅) + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅(b̅ + c̅) + a̅bc + ab̅c̅ + ab̅c + abc̅ => a̅(b̅ + c̅) + a̅bc + ab̅(c̅ + c) + abc̅ => a̅(b̅ + c̅) + a̅bc + ab̅ + abc̅
=> a̅(b̅ + c̅) + a̅bc + a(b̅ + bc̅)
=> a̅(b̅ + c̅) + a̅bc + a(b̅ + c̅) => distribution law. like (a̅b + ab) becomes b(a̅ + a)
=> (b̅ + c̅)(a̅ + a) + a̅bc
=> b̅ + c̅ + a̅bc => absorption through distribution law.
=> b̅ + c̅ + a̅b => absorption through distribution law.
=> b̅ + c̅ + a̅ => Done.
𝒇 = 1
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