gistfile1.txt

    QUESTION 1:
    
    Which of the following formal propositions says that there is no largest prime, where P(x) means "x is prime"? (There may be more than one. You have to select all correct propositions.) The variables denote natural numbers. [6 points]
    
    Math       : ∀x∃y[P(x)∧P(y)∧(x<y)]
    MathEnglish: For all x there exists y such that x and y are both prime, and y is greater than x.
    INCORRECT: not all numbers are prime
    
    English               : There is no largest prime number.
    Slightly mathy English: For all prime numbers, there exists a larger prime number.
    Math                  : ∀x[P(x) => ∃y[P(y) ^ (x<y)] ]
    
    English               : There is no largest prime number.
    Slightly mathy English: For all numbers, there exists a larger prime number.
    Math                  : ∀x∃y[P(y)∧(x<y)]
    
    
    QUESTION 2:
    
    The symbol ∃! means "There exists a unique x such that ...". Which one of the following accurately defines the expression ∃!xϕ(x)? [3 points]
    
    English    : There exists a unique x such that ϕ(x) is true.
    MathEnglish: There exists an x such that ϕ(x) is true and for all y, if y is not equal to x, ϕ(y) is false.
    Alternate  : There exists an x such that ϕ(x) is true and for all y, ϕ(y) implies x equals y.
    Math       :  ∃x[ϕ(x)^ ∀y[ ϕ(y)=> (y = x)]]
    
    
    QUESTION 3:
    
    Which one of the following means "The arithmetic operation x↑y is not commutative"? [2 points]
    
    English    : The arithmetic operation for at least 2 numbers, x and y, x↑y result is not same as the result from y↑x
    MathEnglish: There exists an x and a y such that x↑y is not equal to y↑x.
    Math       : ∃x∃y(x↑y≠y↑x)
    
    
    QUESTION 4:
    
    Which one of the following means "Everybody loves a lover", where L(x,y) means (person) x loves (person) y and a lover is defined to be someone in a mutual loving relationship? (If English is not your native language, you might want to discuss this sentence with a native English speaker before you answer. It's an idiomatic expression.) [5 points]
    
    English: Everybody loves a lover.
                            a lover =  a person who loves someone and that someone loves the former
    MathEnglish: .... For all y and x, if there exists z, such that x loves z and z loves x, then y loves x.
    Math: ∀y∀x[∃z(L(x,z)∧L(z,x)=>L(y,x)]
    

    ∀y∀x∃z[(L(x,z)∧L(z,x)=>L(y,x)]
    MathEnglish: .... For all y and x, there exists z, such that if x loves z and z loves x, then y loves x.
    INCORRECT.

    
    QUESTION 5:
    
    Which of the following statements about the order relation on the real line is/are false? Note that, by convention, ∧ and ∨ have higher precedence than (and hence apply before) ⇒ . [4 points]
    
    Math       : ∀x∀y∀z[(x≤y)∧(y≤z)⇒(x≤z)]
    MathEnglish: For every x, y and z, if x is smaller than or equal to y, and y is smaller than or equal to z, then x is smaller than or equal to z.
    Result     : True.
    
    Math       : ∀x∀y[(x≤y)∧(y≤x)⇒(x=y)]
    MathEnglish: For every x and y, if x is smaller than or equal to y, and y is smaller than or equal to x, then x is equal to y.
    Result     : True.
    
    Math       : ∀x∃y[(x≤y)∧(y≤x)]
    MathEnglish: For all x there exists a y, x is smaller than or equal to y, and y is smaller than or equal to x.
    Result     : True.
    
    Math       : ∃x∀y[(y<x)∨(x<y)]
    MathEnglish: There exists an x such that for any y, y is less than x or x is less than y.
    Result     : False.