Chapter L: Logic
Section L.1-L.2
is the science of correct reasoning and making valid conclusions.
• Conclusions must be inevitable (the conclusion should not change with perspective).
• Every concept must be well defined (there is no scope of ambiguity).
A
but not both.
is a declarative sentence that is either true or false,
Note: Commands, questions, exclamatory and ambiguous (both true and false or noncharacterizable) sentences cannot be statements.
Example 1 Determine which 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 Aggie Soccer team beat Florida Atlantic University 2-0 yesterday.
(f) The number 6 is a prime.
(g) x + 2 = 5
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A statement that expresses a single thought is called a simple statement. We will
denote 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,” that
A
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
T
T
T
F
F
T
F
F
p∧q
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Example 2 Write the compound statement “A&M is the oldest public university in Texas
and it is the largest university in Texas” 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
T
T
T
F
F
T
F
F
p∨q
Example 3 Write the compound statement “Emily is in the Aggie soccer team or she
loves seafood” in symbolic form.
Note: There is a difference in the use of word “or” in everyday language and in mathematical logic. The above mathematical use is also sometimes called the inclusive disjunction.
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An
means
and is written symbolically as
.
The exclusive disjunction p∨q is true if only one of p and q is true. Otherwise, it is false.
Truth table for p∨q
p
q
p∨q
T
T
T
F
F
T
F
F
Example 4 Write the compound statement “Robert is either going to Kyle Field Celebration or he is going to the Hot Air Balloon festival, but not both” in symbolic form.
Note: The exclusive disjunction bears more resemblance to the use of the word “or” in
everyday language as it conveys the meaning “one or the other, but not both”.
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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 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 frosted flakes for breakfast.
Write the following statements in symbolic form.
(a) Either Ted ate french toast and bacon for breakfast or he had frosted flakes but not
bacon for breakfast.
(b) Ted did not eat both french toast and frosted flakes for breakfast but he did have bacon.
(c) Ted ate french toast but did not have bacon or frosted flakes for breakfast.
Example 7 Let p and q denote the following statements:
p: Ted ate french toast for breakfast.
q: Amy ate bacon.
Write the following symbolic forms as compound statements.
(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 symbolic forms.
(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
We say that a statement is a
the 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.
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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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