{{ message }}

Instantly share code, notes, and snippets.

# RobertTalbert/mth225-LO-version1.md

Created Jun 1, 2021

# MTH 225 Learning Objectives

## By module

• Module 1: Arithmetic
• Given an integer in base 2, 8, 10, or 16, represent it using another base.
• Add, subtract, multiply, and divide integers in base 2, 8, and 16.
• Use two's complement to represent a negative integer in binary.
• State the Division Algorithm and use it to find the quotient and remainder when dividing one positive integer by another.
• Compute `a % b` for an integer `a` and positive integer `b`.
• Module 2: Logic
• Differentiate between a proposition and a statement that isn't a proposition.
• Determine the truth value of a compound proposition formed by using `AND`, `OR`, or `NOT`.
• Identify the hypothesis and conclusion of a conditional statement.
• State the conditions under which a conditional statement is true.
• Given a conditional statement and the truth values of its hypothesis and conclusion, state whether the conditional statement is true or false.
• State the converse and contrapositive of a conditional statement.
• Construct a truth table for `AND`, `OR`, `NOT`, and conditional statements.
• Construct a truth table for a compound statement involving more than one logical connective and three or more variables.
• Determine whether two propositions are logically equivalent by using a truth table.
• Explain the differences between a predicate and a proposition.
• Determine the domain and truth set of a predicate.
• Determine the truth value of existentially and universally quantified predicates, and determine if a predicate is underdetermined.
• Determine the truth value of a double-quantified predicate.
• State the negation of a quantified predicate.
• Module 3: Sets and Functions
• Determine whether a set is finite or infinite.
• Identify "famous" sets corresponding to their notation: The empty set, the natural numbers, the whole numbers, the integers, and the rational, real, and complex numbers.
• Given a set in roster form, rephrase the set in set-builder notation; and vice versa.
• Given a set in either roster or set-builder form and an object, determine whether the object is a member of the set.
• State the definition of "subset" and given two sets, determine if one is a subset of the other.
• State the definition of "equality" of sets, and given two sets, determine if they are equal.
• Given a finite set, write its power set.
• Given a finite set, state its cardinality.
• Given two sets, find their intersection, union, difference, symmetric difference, and Cartesian product.
• Given a set and its universal set, find its complement.
• State the definition of a function between two sets.
• Given a mapping between two sets, determine whether or not it is a function. If it is, state the domain, range, and codomain.
• State the definitions of injective, surjective, and bijective functions.
• Given a function between two sets, determine whether it is injective, surjective, or bijective.
• Compute outputs using the floor, ceiling, factorial, `DIV`, and `MOD` functions.
• Module 4: Combinatorics
• State the Additive Principle and use it to count the number of arrangements in an appropriate setting.
• State the Multiplicative Principle and use it to count the number of arrangements in an appropriate setting.
• Define the binomial coefficient in terms of the solution to a counting problem.
• State the recurrence relation that defines the binomial coefficient.
• State the closed formula for the binomial coefficient \$\binom{n}{k}\$
• Compute values of the binomial coefficient using recursion, Pascal's Triangle, and the closed formula.
• Count the number of permutations of a group of objects and the number of \$k\$-permutations from a set of \$n\$ objects.
• Use the "stars and bars" method to count the number of ways to distribute objects among a group.
• State the Principle of Inclusion and Exclusion and use it to count complex arrangements.
• Given a counting problem, identify which method is most appropriate which are not appropriate.
• Module 5: Sequences, Recursion, and Induction
• Compute values in a sequence defined as a closed formula.
• Compute values in a sequence defined as a recurrence relation.
• Given a finite sum, rewrite it using sigma notation; given sigma notation, calculate the sum.
• Given a sequence, identify it as arithmetic or geometric; then find closed-form and recursive expressions for the sequence.
• Determine a recurrence relation for a given recursive sequence.
• Given a proposed solution for a recurrence relation, prove or disprove that it solves the relation.
• Use the characteristic roots method to solve a second-order linear homogeneous recurrence relation.
• State the Principle of Mathematical Induction and explain in basic terms how and why it works.
• Set up the framework for a mathematical proof using Mathematical Induction:
• Identify the predicate
• Establish the base case
• State the inductive hypothesis
• State what should be proven in the inductive step and outline how the proof would unfold
• Perform a critical analysis of a written proof by mathematical induction.
• Construct a clear and correct mathematical proof using mathematical induction.