c
Math 166, Fall 2011, Benjamin
Aurispa
Chapter L: Logic
L.1: Introduction to Logic and L.2: Truth Tables
A statement, or proposition, is a declarative sentence that can be classified as either true or false, but
not both.
Example: Determine whether the following are statements or not.
• Ben teaches math at A&M.
• Get out of here!
• Austin is 7.5 million inches from College Station.
• What is your name?
• Ben teaches history at A&M.
• 2+2=3
• x − 12 = 2
• x2 ≥ 0 for all real numbers x.
Statements that express a single complete thought are called simple statements. We use the lowercase letters
p, q, r, etc. to represent simple statements.
We can combine 2 or more simple statements to form compound statements. We often want to determine
when a compound statement is true or false. In other words, we want to determine the truth value of
the proposition. We can use truth tables to list all the possible combinations of truth values for the
proposition.
The words that are used to combine statements are called connectives. There are four connectives we’ll
use:
1. Conjunction
Connective: “and” Symbol: ∧
A conjunction is a proposition of the form “p and q” and is written symbolically: p ∧ q
A conjunction p ∧ q is true if
Truth table for p ∧ q:
p
q
T
T
T
F
F
T
F
F
Example: Let p and q denote the propositions:
p: Ben is 28 years old.
q: Ben lives in Bryan, TX.
State the compound proposition p ∧ q in words:
1
c
Math 166, Fall 2011, Benjamin
Aurispa
2. Inclusive Disjunction
Logical Connective: “or” Symbol: ∨
An inclusive disjunction is a proposition of the form “p or q” and is written symbolically as: p ∨ q
An inclusive disjunction p ∨ q is true if
Truth table for p ∨ q:
p
q
T
T
T
F
F
T
F
F
Example: Let p and q denote the propositions:
p: The number of males at A&M increased in 2010.
q. The number of females at A&M increased in 2010.
State the proposition p ∨ q in words:
3. Exclusive Disjunction
Logical Connective: “or” Symbol: Y
An exclusive disjunction is a proposition of the form “either p or q” and is written: p Y q
The exclusive disjunction p Y q is true if
Truth table for p Y q:
p
q
T
T
T
F
F
T
F
F
Example: Let p and q denote the propositions:
p: A burger deal at Sonic comes with french fries.
q. A burger deal at Sonic comes with tater tots.
State the proposition p Y q in words:
In mathematics, the word “or” is used in the inclusive sense unless otherwise specified. To distinguish
the exclusive disjunction from the inclusive injunction, use the words “either...or.”
2
c
Math 166, Fall 2011, Benjamin
Aurispa
4. Negation
Logical Connective: “not” Symbol: ∼
A negation is a proposition of the form “not p” and is written: ∼ p
The negation ∼ p is true if
Truth table for ∼ p:
p
T
F
Example: Let p, q, r, and s denote the propositions:
p: The price of gas increased in 2007.
q: Some students at Texas A&M live in the dorms.
r: All Aggies are from Texas.
s: No elephants weigh less than 2000 pounds.
Write the negations for each of these statements:
∼p
∼q
∼r
∼s
Example: Let p, q, and r denote the following propositions:
p: The number of immigrants in Texas declined last year.
q: The population of Texas did not rise last year.
r: The number of schools in Texas increased last year.
State the following compound propositions in words:
• ∼pYq
• ∼q ∧∼r
• p ∨ (∼ q ∧ r)
3
c
Math 166, Fall 2011, Benjamin
Aurispa
Example: Let p, q, and r denote the following propositions:
p: Jack ate a hamburger at lunch.
q: Jack ate french fries at lunch.
r: Jack had a Dr. Pepper at lunch.
Express the following statements symbolically.
• Either Jack ate a hamburger and fries at lunch or he had a Dr. Pepper but not fries at lunch.
• Jack did not eat both a hamburger and a Dr. Pepper at lunch, but he did have fries.
• Jack ate a hamburger but did not eat fries or have a Dr. Pepper at lunch.
Examples: Construct truth tables for the following compound propositions.
• ∼ (p ∨ ∼ q)
p
q
T
T
T
F
F
T
F
F
• (p ∨ q) Y (∼ p ∧ q)
p
q
T
T
T
F
F
T
F
F
4
c
Math 166, Fall 2011, Benjamin
Aurispa
• ∼ (p ∧ q)∨ ∼ r
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
• (∼ p Y q) ∧ (q ∨ r)
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
If a compound statement is FALSE no matter what the original truth values of the simple statements are,
we call this a contradiction.
If a compound statement is TRUE no matter what the original truth values of the simple statements are,
we call this a tautology.
Example: Consider the statements:
p: Florida is an island. and q: An apple is a type of fruit.
Note that p is false and q is true. Determine the truth values of the following compound statements.
∼p∧q
∼ (p Y q)
5
c
Math 166, Fall 2011, Benjamin
Aurispa
Additional Honors Topic: Conditional and Biconditional Connectives
“If...then” statements are very common in the English language.
5. Conditional Statements
Connective: “if...then” Symbol: →
A conditional statement is a proposition of the form “if p, then q” and is written symbolically as: p → q
p is called the hypothesis; q is called the conclusion.
Truth table for p → q:
p
q
T
T
T
F
F
T
F
F
Example: Let p and q denote the propositions:
p: Billy makes an A on his spelling test.
q: Billy goes to McDonald’s.
State the compound proposition p → q in words:
A conditional statement is only considered FALSE when the “promise” of the “if...then” is BROKEN. In
other words, when the hypothesis is true and the conclusion is false. In all other cases, it is considered true.
5. Biconditional Statements
Connective: “if and only if” Symbol: ↔
A biconditional statement is a proposition of the form “p if and only if q” and is written symbolically
as: p ↔ q
Truth table for p ↔ q:
p
q
T
T
T
F
F
T
F
F
Example: Let p and q denote the propositions:
p: Jack eats at Taco Bell.
q: Jill eats at Taco Bell.
State the compound proposition p ↔ q in words:
A biconditional statement is true when both p and q are true or when both p and q are false.
6
c
Math 166, Fall 2011, Benjamin
Aurispa
Example: Let p, q, and r denote the following propositions:
p: Salaries increase at AggieCorp.
q: Benefits do not rise at AggieCorp.
r: Layoffs occur at AggieCorp.
State the following compound propositions in words:
• ∼p→q
• r ↔ (p ∧ q)
• (p∨ ∼ q) → ∼ r
Examples: Construct truth tables for the following compound propositions. ∼ p → (p Y q)
p
q
T
T
T
F
F
T
F
F
(p ∧ r) ↔ (q ∨ r)
p
q
r
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
7