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Math 166, Fall 2015, Robert
Williams
Chapter L- Logic
Statements
A statement (or proposition) is a
that is ei-
ther
.
Some types of sentences that are not statements:
• Questions
• Commands
• Exclamations
• Ambiguous sentences
Example 1 Determine if each of the following sentences is a statement. If it
is a statement, mark it with an “S”. Otherwise, mark it with an “N”.
• Math 166 begins with a discussion on logic.
• 2∗3=6
• The stars are so beautiful!
• Every student in this class will make an A.
• How long does it take to drive to Houston?
• The sun is one million feet wide.
• This class lasts for 3 hours every day.
• x2 > 0 whenever x > 0.
• 1+1=3
• x−3=5
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Math 166, Fall 2015, Robert
Williams
A statement is called a simple statement if it expresses a single thought. We
often use lower case letters (p, q, r, etc.) to represent simple statements. All of
the statements in Example 1 are examples of simple statements. A compound
statement is a statement that expresses multiple thoughts such as “I will eat a
snack or have an early dinner.”
Connectives and Truth Tables
A connective is a word or words that are used to combine multiple simple
statements into a single compound statement. We often want to deteremine
whether or not a compound statement is true, that is to determine its truth
value. A helpful aid in doing so is to construct a truth table where we list all
possible combinations of truth values for our statement. We will focus on the
following four types of connectives:
A negation is a statement of the form “not p” and is written as ∼ p.
The statement ∼ p is true when
p
∼p
T
F
A conjunction is a statement of the form “p and q” and is written as p ∧ q.
The statement p ∧ q is true when
p
q
T
T
T
F
F
T
F
F
p∧q
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Math 166, Fall 2015, Robert
Williams
A disjunction (sometimes called an inclusive disjunction) is a statement of
the form “p or q” and is written as p ∨ q. The statement p ∨ q is true when
p
q
T
T
T
F
F
T
F
F
p∨q
An exclusive disjunction is a statement of the form “p or q, but not both”
and is written as pYq. The statement pYq is true when
p
q
T
T
T
F
F
T
F
F
pYq
Important: In math, the word “or” is always the inclusive “or”. That is, it is
always an inclusive disjunction.
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Math 166, Fall 2015, Robert
Williams
Example 2 Let p and q be the following statements:
p: I woke up at 6:00 AM
q: My first class was at 9:30 AM
Write the following statements in symbolic form:
1. I woke up at 6:00 AM or my first class was at 9:30 AM
2. I woke up at 6:00 AM, and my first class is not at 9:30 AM
3. I did not wake up at 6:00 AM or my first class was not at 9:30 AM.
Example 3 Let p, q, and r be the following statements:
p: All students at A& M live in dorms.
q: Everyone in this class owns a laptop.
r: More students have cats than dogs.
Write the following statements:
1. ∼ p
2. p Y r
3. ∼ p ∧ q
4. r∧ ∼ (p ∨ q)
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Math 166, Fall 2015, Robert
Williams
Example 4 Construct a truth table for the following compound propositions
• ∼ p ∨ (p ∧ q)
p
q
T
T
T
F
F
T
F
F
• ∼ (p ∧ q) ∨ (p Y q)
p
q
T
T
T
F
F
T
F
F
• (p ∧ q) ∨ (p ∧ 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
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Math 166, Fall 2015, Robert
Williams
• (∼ p ∨ 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
A statement is called a contradiction if the truth value of the statement is
False no matter what the truth values of its component simple statements are.
A statement is a tautology if the truth value of the statement is True no matter
what the truth value of its component simple statements are.
Give an example of a contradcition:
Give an example of a tautology:
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