1. 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.

2. Conjunction Truth Table
3.2 – Truth Tables and Equivalent Statements
Conjunction Truth Table
p and q
p q
p ˄ q
T T T
T F F
F T
F F

3. Finding the Truth Value of a Conjunction
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
p q
p ˄ q
T T T
T F F
F T
F F
12 < 9
q is false
The truth value for p ˄ q is false

4. 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
F T
F F F

5. Finding the Truth Value of a Disjunction
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
T T T
T F
F T
F F F
12 < 9
q is false
The truth value for p ˅ q is true

6. 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

7. Example: Constructing a Truth Table
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
~ p
~ q
~ p ˅ ~ q
p ˄ (~ p ˅ ~ q)
T T
T F
F T
F F

8. Example: Constructing a Truth Table
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
~ p
~ q
~ p ˅ ~ q
p ˄ (~ p ˅ ~ q)
T T F
T F
F T T
F F

9. Example: Constructing a Truth Table
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
~ p
~ q
~ p ˅ ~ q
p ˄ (~ p ˅ ~ q)
T T F
T F T
F T
F F

10. Example: Constructing a Truth Table
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
~ p
~ q
~ p ˅ ~ q
p ˄ (~ p ˅ ~ q)
T T F
T F T
F T
F F

11. Example: Constructing a Truth Table
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
~ p
~ q
~ p ˅ ~ q
p ˄ (~ p ˅ ~ q)
T T F
T F T
F T
F F

12. Example: Mathematical Statements
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
̴ p ̴ q
T T
T F
F T
F F
̴ p ˄ ̴ q

13. Example: Mathematical Statements
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
̴ p ̴ q
T T
F F
T F
F T
̴ p ˄ ̴ q

14. Example: Mathematical Statements
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
̴ p ̴ q
T T
F F
T F
F T
̴ p ˄ ̴ q F T
The truth value for the statement is false.

15. Example: Mathematical Statements
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
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 ˅

16. Example: Mathematical Statements
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 ˅

17. Example: Mathematical Statements
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 F T ˅

18. Example: Mathematical Statements
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 F T ˅

19. Example: Mathematical Statements
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 F T ˅ F T
The truth value for the statement is true.

20. Equivalent Statements
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 q
~ p ˄ ~ q
̴ (p ˅ q)
T T
T F
F T
F F

21. Equivalent Statements
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 q
~ p ˄ ~ q
̴ (p ˅ q)
T T F
T F
F T
F F T

22. Equivalent Statements
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 q
~ p ˄ ~ q
̴ (p ˅ q)
T T F
T F
F T
F F T
Yes