MTH 231 Study Guide - Final Guide: Logical Biconditional, Empty Set, Commutative Property
7 Jun 2018
School
Department
Course
Professor
Oregon State
University
MTH 231
Elements of Discrete Mathematics
Spring 2018
Final Exam
Professor Charlie Robson
Exam Guide
Topics Included:
Section 1.1: Propositional Logic
find more resources at oneclass.com
find more resources at oneclass.com
Section 1.3: Propositional Equivalences
Section 1.4: Predicates and Quantifiers
Section 1.7: Intro to Proofs
Section 1.8: Proof Methods & Strategy
Section 2.1:Sets
Section 2.2: Cartesian Product
Section 5.1: Mathematical Introduction
Section 5.2: Strong P.M.I
Section 6.1: The Basis of Counting
Section 6.3: Permutations and Combinations
Section 6.4: Binomial Coefficient and Identities
Section 1.1: Propositional Logic
Propositions
A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either
find more resources at oneclass.com
find more resources at oneclass.com
true
or false, but not both.
EXAMPLE 1 All the following declarative sentences are propositions.
1. Washington, D.C., is the capital of the United States of America.
2. Toronto is the capital of Canada.
3. 1 + 1 = 2.
4. 2 + 2 = 3.
Solutions:
● Propositions 1 and 3 are true, whereas 2 and 4 are false. ▲
● Some sentences that are not propositions are given in Example 2.
EXAMPLE 2 Consider the following sentences.
1. What time is it?
2. Read this carefully.
3. x + 1 = 2.
4. x + y = z.
Solutions:
● Sentences 1 and 2 are not propositions because they are not declarative sentences.
● Sentences 3 and 4 are not propositions because they are neither true nor false.
○ Note that each of sentences 3 and 4 can be turned into a proposition if we assign
values to the variables.
EXAMPLE 3 Find the negation of the proposition----“Michael’s PC runs Linux” and express this
in simple English.
Solution:
● The negation is “It is not the case that Michael’s PC runs Linux.”
● This negation can be more simply expressed as “Michael’s PC does not run Linux.”
EXAMPLE 4 Let p be the statement Maria learns discrete mathematics and q the statement
Maria will find a good job. Express the statement p → q as a statement in English.
Solution:
● From the definition of conditional statements, we see that when p is the statement
“Maria learns discrete mathematics” and q is the statement “Maria will find a good job,”
p → q represents the statement
● “If Maria learns discrete mathematics, then she will find a good job.”
There are many other ways to express this conditional statement in English. Among the
most natural of these are:
○ “Maria will find a good job when she learns discrete mathematics.”
○ “For Maria to get a good job, it is sufficient for her to learn discrete mathematics.”
○ “Maria will find a good job unless she does not learn discrete mathematics.”
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
A proposition is a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both. Example 1 all the following declarative sentences are propositions: washington, d. c. , is the capital of the united states of america, toronto is the capital of canada, 1 + 1 = 2, 2 + 2 = 3. Propositions 1 and 3 are true, whereas 2 and 4 are false. Some sentences that are not propositions are given in example 2. Example 2 consider the following sentences: what time is it, read this carefully, x + 1 = 2, x + y = z. Sentences 1 and 2 are not propositions because they are not declarative sentences. Sentences 3 and 4 are not propositions because they are neither true nor false. Note that each of sentences 3 and 4 can be turned into a proposition if we assign values to the variables.
