Chapter L: Logic
Section L.1-L.2
is the science of correct reasoning and making valid conclusions.
• Conclusions must be inescapable (the conclusion cannot change from perspective
to perspective)
• Every concept must be clearly defined (no ambiguity in meaning)
A
but not both.
is a declarative sentence that is either true or false,
Note: Commands, questions, exclamations, and ambiguous sentences cannot be statements.
Example 1 Determine whether each of the following are statements. If a sentence is not
a statement, explain why not.
a) Sit down.
b) Are you going out to eat tonight?
c) What a beautiful picture!
d) This class is great.
e) The Humbugs ate cheesecake for breakfast
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f) Two times three equals five.
A statement that represents a single thought is called a simple statement. We will
represent a simple statement by a lowercase letter, such as p or q.
Example: I am in Math 166.
A compound statement is formed when two or more simple statements are connected.
Example: I am in Math 166 and I am learning about logic.
The truth value of a statement can be either true or false. Logic does not concern
itself with whether simple statements are true or false, but if the truth values of all the
simple statements that make up a compound statement are known, logic can determine
the truth value of a compound statement.
is a word or words, such as “and” or “if and only if,”
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that is used to combine two or more simple statements into a compound statement.
Types of Connectives:
1. Conjunction
2. Disjunction
3. Exclusive Disjunction
4. Negation
A
is a statement of the form
and is written symbolically as
.
The conjunction p ∧ q is true if both p and q are true. Otherwise, it is false.
Truth Table for p ∧ q
p q p∧q
T T
T F
F T
F F
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Example 2 Write the compound statement “Blue eats fish and Tonks is a cat” in symbolic
form.
A
is a statement of the form
and is written symbolically as
.
The disjunction p ∨ q is false if both p and q are false. Otherwise, it is true.
Truth Table for p ∨ q
p q p∨q
T T
T F
F T
F F
Example 3 Write the compound statement “The Humbugs bought a house or they went
to Madagascar” in symbolic form.
An
means
and is written symbolically as
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.
Truth Table for p∨q
p q p∨q
T T
T F
F T
F F
Example 4 Write the compound statement “Cindy either went to the Blue Man Group
concert or she went to the company party, but not both” in symbolic form.
A
is a statement of the form
and is written symbolically as
.
The negation ∼ p is true if p is false and false if p is true.
Truth Table for ∼ p
p ∼p
T
F
Example 5 Write the compound statement “Cindy did not go to the company party” in
symbolic form.
Order of Precedence The logical connectives are used in the following order: ∼, ∧, ∨.
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Example 6 Let p, q, and r denote the following statements:
p: Ted ate French Toast for breakfast.
q: Ted ate bacon for breakfast.
r: Ted had whipped cream for breakfast.
Write the following statements in symbolic form.
a) Either Ted ate French Toast and bacon for breakfast or he had whipped cream but not
bacon for breakfast.
b) Ted did not eat both French Toast and whipped cream for breakfast, but he did have
bacon.
c) Ted ate French Toast but did not have bacon or whipped cream for breakfast.
Example 7 Let p and q denote the following statements:
p: Ted ate French Toast for breakfast.
q: Amy ate bacon.
Write out the statements that correspond to each of the following.
a) p ∧ q
b) p ∨ q
c) p∨q
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d) p∨ ∼ q
Example 8 Let p, q, and r denote the following statements:
p: The number of people who moved to Texas declined last year.
q: The population of Texas did not rise last year.
r: The number of schools in Texas increased last year.
Write out the statements that correspond to each of the following.
a) ∼ p∨q
b) ∼ q∧ ∼ p
c) p ∨ (∼ q ∧ r)
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Example 9 Construct a truth table for each of the following statements.
a) ∼ (p∨ ∼ q)
b) (p ∨ q)∨(∼ p ∧ q)
c) ∼ (p ∧ q)∨ ∼ r
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We say that a statement is a
if the truth value of the
statement is always false no matter what the truth values of its simple component statements.
Example p∧ ∼ p is a contradiction.
if the truth value of the
We say that a statement is a
statement is always true no matter what the truth values of its simple component statements.
Example p∨ ∼ p is a tautology.
Example 10 Determine if the statement (p∨ ∼ q)∨(∼ p∧q) is a tautology, contradiction,
or neither
Example 11 Let p and q be the following statements:
p: Alaska is an island.
q: Michigan is a peninsula.
Determine the truth value of each of the following statements. (Note that p is false and
q is true.)
a) ∼ p ∧ q
b) ∼ (p∨q)
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