# MA121 Study Guide - Midterm Guide: Distributive Property, Disjoint Sets, Symmetric Difference

182 views40 pages

28 Feb 2018

School

Department

Course

Professor

For unlimited access to Study Guides, a Grade+ subscription is required.

Wilfrid Laurier University

MA121

Introduction to Mathematical Proofs

Fall 2017

Term Test 2

Prof: Robert John Rundle

Exam Guide

Topics Covered

CHAPTER 3: SET THEORY

3.1 LOGIC VERSUS SET THEORY

3.2 SYMMETRIC DIFFERENCE

3.3 ALGEBRA OF SETS AND THE DUALITY PRINCIPLE

3.4 FINITE SETS, POWER SETS AND COUNTING PRINCIPLES

3.5 ORDERED PAIRS AND CARTESIAN PRODUCTS

3.6 FUNCTIONS AND THE PIGEONHOLE PRINCIPLE

CHAPTER 4: COMBINATORICS

4.1 PERMUTATIONS AND COMBINATIONS

4.2 BINOMIAL COEFFICIENTS

4.3 PASCAL'S TRIANGLE

4.4 THE BINOMIAL THEOREM

CHAPTER 5: NUMBER THEORY

5.2 DIVISIBILITY

5.2 DIVISION ALGORITHM

5.2 GREATEST COMMON DIVISOR

5.2 EUCLID’S ALGORITHM

5.2 FUNDAMENTAL THEOREM OF ARITHMETIC

CHAPTER 3: SET THEORY

3.1 LOGIC VERSUS SET THEORY

Definition: Logic is the study of deductive reasoning, the process of using mathematical

sentences to make decisions.

Let

QP,

be set and statements

qp,

as

QxPx ∈∈ ,

Then

qp ∧

is the same as

QP ∩

qp ∨

is the same as

QP ∪

p~

is the same as

c

P

Recall set

Theorem: Let X be an arbitrary set and let P(X) be the set of all subsets of X. P(X) is called the

power set of X. Let A, B, and C be arbitrary elements of P(X).

a. A ⋂ B = B ⋂ A (Commutative Law for Intersection)

A ⋃ B = B ⋃ A (Commutative Law for Union)

b. A ⋂ (B ⋂ C) = (A ⋂ B) ⋂ C (Associative Law for Intersection)

A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ C (Associative Law for Union)

c. A ⋂ B ⊂ A

d. A ⋂ X = A; A ⋃ ∅ = A

e. A ⊂ A ⋃ B

f. A ⋃ X = X; A ⋂ ∅ = ∅

g. A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C) (Distributive Law of Union with respect to Intersection)

A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C) (Distributive Law of Intersection with respect to Union)

h.

∁

(A ⋃ B) =

∁

A ⋂

∁

B

∁

(A ⋂ B) =

∁

A ⋃

∁

B

i. A ⋂

∁

A = ∅; A ⋃

∁

A = X