conjunctions the truth values of component statements are used to find the truth values of compound...
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Conjunctions
The truth values of component statements are used to find the truth values of compound statements.
The truth values of the conjunction p and q (p ˄ q), are given in the truth table on the next slide. The connective “and” implies “both.”
3.2 – Truth Tables and Equivalent StatementsTruth Values
Truth Table
A truth table shows all four possible combinations of truth values for component statements.
Conjunction Truth Table
p q p ˄ q
T T T
T F F
F T F
F F F
p and q
3.2 – Truth Tables and Equivalent Statements
Finding the Truth Value of a Conjunction
If p represent the statement 4 > 1 and q represent the statement 12 < 9, find the truth value of p ˄ q.
3.2 – Truth Tables and Equivalent Statements
p q p ˄ q
T T T
T F F
F T F
F F F
p and q
12 < 9
4 > 1 p is true
q is false
The truth value for p ˄ q is false
Disjunctions
The truth values of the disjunction p or q (p ˅ q) are given in the truth table below. The connective “or” implies “either.”
3.2 – Truth Tables and Equivalent Statements
p q p ˅ q
T T T
T F T
F T T
F F F
p or q
Disjunction Truth Table
Finding the Truth Value of a Disjunction
If p represent the statement 4 > 1, and q represent the statement 12 < 9, find the truth value of p ˅ q.
3.2 – Truth Tables and Equivalent Statements
p q p ˅ q
T T T
T F T
F T T
F F F
p or q
12 < 9
4 > 1 p is true
q is false
The truth value for p ˅ q is true
Negation
The truth values of the negation of p ( 2 p) are given in the truth table below.
p 2̴ p
T F
F T
not p
3.2 – Truth Tables and Equivalent Statements
Example: Constructing a Truth Table
p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q)
T T
T F
F T
F F
Construct the truth table for: p ˄ (~ p ˅ ~ q)
3.2 – Truth Tables and Equivalent Statements
A logical statement having n component statements will have 2n rows in its truth table.
22 = 4 rows
Example: Constructing a Truth Table
p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q)
T T F
T F F
F T T
F F T
Construct the truth table for: p ˄ (~ p ˅ ~ q)
3.2 – Truth Tables and Equivalent Statements
A logical statement having n component statements will have 2n rows in its truth table.
22 = 4 rows
Example: Constructing a Truth Table
p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q)
T T F F
T F F T
F T T F
F F T T
Construct the truth table for: p ˄ (~ p ˅ ~ q)
3.2 – Truth Tables and Equivalent Statements
A logical statement having n component statements will have 2n rows in its truth table.
22 = 4 rows
Example: Constructing a Truth Table
p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q)
T T F F F
T F F T T
F T T F T
F F T T T
Construct the truth table for: p ˄ (~ p ˅ ~ q)
3.2 – Truth Tables and Equivalent Statements
A logical statement having n component statements will have 2n rows in its truth table.
22 = 4 rows
Example: Constructing a Truth Table
p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q)
T T F F F F
T F F T T T
F T T F T F
F F T T T F
Construct the truth table for: p ˄ (~ p ˅ ~ q)
3.2 – Truth Tables and Equivalent Statements
A logical statement having n component statements will have 2n rows in its truth table.
22 = 4 rows
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q ̴� p 2 q
T T
T F
F T
F F
� p ˄ 2 q�̴ p ˄ 2 q
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q ̴� p 2 q
T T F F
T F F T
F T T F
F F T T
� p ˄ 2 q�̴ p ˄ 2 q
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q ̴� p 2 q
T T F F
T F F T
F T T F
F F T T
� p ˄ 2 q
F
F
F
T
�̴ p ˄ 2 q
The truth value for the statement is false.
˅
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q r ̴� p 2 q 2 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 ˄ r � q ˄ p( � p ˄ r) ˅ ( � q ˄ p)
˅
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q r ̴� p 2 q 2 r
T T T
F F F
T T F F F T
T F T F T F
T F F F T T
F T T T F F
F T F T F T
F F T T T F
F F F T T T
� p ˄ r � q ˄ p( � p ˄ r) ˅ ( � q ˄ p)
˅
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q r ̴� p 2 q 2 r
T T T
F F F
T T F F F T
T F T F T F
T F F F T T
F T T T F F
F T F T F T
F F T T T F
F F F T T T
� p ˄ r � q ˄ p
F
F
F
F
T
F
T
F
( � p ˄ r) ˅ ( � q ˄ p)
˅
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q r ̴� p 2 q 2 r
T T T
F F F
T T F F F T
T F T F T F
T F F F T T
F T T T F F
F T F T F T
F F T T T F
F F F T T T
� p ˄ r � q ˄ p
F F
F F
F T
F T
T F
F F
T F
F F
( � p ˄ r) ˅ ( � q ˄ p)
Example: Mathematical Statements
If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false.
3.2 – Truth Tables and Equivalent Statements
p q r ̴� p 2 q 2 r
T T T
F F F
T T F F F T
T F T F T F
T F F F T T
F T T T F F
F T F T F T
F F T T T F
F F F T T T
� p ˄ r � q ˄ p
F F
F F
F T
F T
T F
F F
T F
F F
( � p ˄ r) ˅ ( � q ˄ p) ˅
F
F
T
T
T
F
T
F
The truth value for the statement is true.
Equivalent Statements
Are the following statements equivalent?
p q ~ p ˄ ~ q � (p ˅ q)
T T
T F
F T
F F
Two statements are equivalent if they have the same truth value in every possible situation.
3.2 – Truth Tables and Equivalent Statements
~ p ˄ ~ q and � (p ˅ q)
Equivalent Statements
Are the following statements equivalent?
p q ~ p ˄ ~ q � (p ˅ q)
T T F
T F F
F T F
F F T
Two statements are equivalent if they have the same truth value in every possible situation.
3.2 – Truth Tables and Equivalent Statements
~ p ˄ ~ q and � (p ˅ q)
Equivalent Statements
Are the following statements equivalent?
p q ~ p ˄ ~ q � (p ˅ q)
T T F F
T F F F
F T F F
F F T T
Yes
Two statements are equivalent if they have the same truth value in every possible situation.
3.2 – Truth Tables and Equivalent Statements
~ p ˄ ~ q and � (p ˅ q)