Transcript 3.2 – Truth Tables and Equivalent Statements Truth Values The truth values of component statements are used to find the truth values of.

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