mth225-LO-version1.md

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.