What is a truth table?

A truth table is a systematic representation of all possible truth values of a logical formula. It shows all possible combinations of values (true/false) of the propositional variables and the final result of the formula for each combination.

How do you know if a formula is a tautology, contradiction, or contingency?

A formula is a TAUTOLOGY if its final column has all true values (1). It is a CONTRADICTION if all values are false (0). It is a CONTINGENCY if it has a mix of true and false values. Tautologies are always logical truths and satisfiable.

How many rows does a truth table have?

The number of rows is 2^n, where n is the number of propositional variables. For example: 2 variables = 4 rows, 3 variables = 8 rows, 4 variables = 16 rows.

What does it mean for a formula to be satisfiable?

A formula is satisfiable if there exists at least one interpretation (combination of values) that makes it true. Tautologies and contingencies are satisfiable. Only contradictions are unsatisfiable.

Truth Tables - All Solved Exercises

Using the truth table method, determine whether
these formulas are (1) tautological, contingent, or contradictory (2) Logical truth or not (3) Satisfiable or not:

1. [Exercise 1] p ∧ q → p

Solution

p	q	p∧q	p ∧ q → p
1	1	1	1
1	0	0	1
0	1	0	1
0	0	0	1

TAUTOLOGY, SATISFIABLE, AND LOGICAL TRUTH

2. [Exercise 2] p ∨ p → r

Solution

p	r	p ∨ p	p ∨ p → r
1	1	1	1
1	0	1	0
0	1	0	1
0	0	0	1

CONTINGENT, SATISFIABLE, NOT A LOGICAL TRUTH

3.p ∨ (q → r)

4.(p → q) ∧ (q → r) → (p → r)

Solution

p	q	r	p → q	q → r	(p → q) ∧ (q → r)	p → r	(p → q) ∧ (q → r) → (p → r)
1	1	1	1	1	1	1	1
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	1
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	1
0	1	0	1	0	0	1	1
0	0	1	1	1	1	1	1
0	0	0	1	1	1	1	1

TAUTOLOGY, SATISFIABLE, AND LOGICAL TRUTH

5. [Exercise 5]p → (q → r)
6. [Exercise 6]p ∨ q → (r ∨ s → p)
7. [Exercise 7]p ∧ q → q ∧ p

Solution

p	q	p ∧ q	q ∧ p	p ∧ q → q ∧ p
1	1	1	1	1
1	0	0	0	1
0	1	0	0	1
0	0	0	0	1

TAUTOLOGY, SATISFIABLE, LOGICAL TRUTH

8. [Exercise 8](p → q) ∧ p → q
9. [Exercise 9](p → q) ∧ p ∧ ¬q
10. [Exercise 10](p → q) ∧ (p → q)

Solution

p	q	p → q	(p → q) ∧ (p → q)
1	1	1	1
1	0	0	0
0	1	1	1
0	0	1	1

CONTINGENT, SATISFIABLE, NOT A LOGICAL TRUTH

11. [Exercise 11](p → q) ∧ q → p
12. [Exercise 12](p → q) ∧ ¬q → ¬p

Solution

p	q	¬p	¬q	p → q	p → q ∧ ¬q	(p → q) ∧ ¬q → ¬p
1	1	0	0	1	0	1
1	0	0	1	0	0	1
0	1	1	0	1	0	1
0	0	1	1	1	1	1

TAUTOLOGY, SATISFIABLE, AND LOGICAL TRUTH

13. [Exercise 13](p → q) ∧ ¬p → ¬q
14. [Exercise 14]¬(p ∧ q) ↔ ¬p ∧ ¬q

Solution

p	q	¬p	¬q	¬p ∧ ¬q	p ∧ q	¬(p ∧ q)	¬(p ∧ q) ↔ ¬p ∧ ¬q
1	1	0	0	0	1	0	1
1	0	0	1	0	0	1	0
0	1	1	0	0	0	1	0
0	0	1	1	1	0	1	1

CONTINGENT, SATISFIABLE, AND NOT A LOGICAL TRUTH

15. [Exercise 15]¬(p ∧ q) ↔ ¬p ∨ ¬q
16. [Exercise 16][(p → q) ∧ (q → r)] ∧ ¬(p → r)

Solution

p	q	r	p → q	q → r	p → q ∧ q → r	p → r	¬(p → r)
1	1	1	1	1	1	1	0
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	0
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	0
0	1	0	1	0	0	1	0
0	0	1	1	1	1	1	0
0	0	0	1	1	1	1	0

CONTRADICTION, UNSATISFIABLE, AND NOT A LOGICAL TRUTH.

17. [Exercise 17]p → (q ∧ ¬r) → ¬q)
18. [Exercise 18]¬(p ∨ q) ↔ ¬r ∨ ¬q

Solution

p	q	r	¬r	¬q	p ∨ q	¬(p ∨ q)	¬r ∨ ¬q	¬(p ∨ q) ↔ ¬r ∨ ¬q
1	1	1	0	0	1	0	0	1
1	1	0	1	0	1	0	1	0
1	0	1	0	1	1	0	1	0
1	0	0	1	1	1	0	1	0
0	1	1	0	0	1	0	0	1
0	1	0	1	0	1	0	1	0
0	0	1	0	1	0	1	1	1
0	0	0	1	1	0	1	1	1

CONTINGENT, SATISFIABLE, AND NOT A LOGICAL TRUTH.

19. [Exercise 19]¬(p ∨ q) ↔ ¬p ∨ ¬r
20. [Exercise 20] ¬(p → q) ↔ (p ∧ r)

Solution

p	q	r	p ∧ r	p → q	¬(p → q)	¬(p → q) ↔ p ∧ r
1	1	1	1	1	0	0
1	1	0	0	1	0	1
1	0	1	1	0	1	1
1	0	0	0	0	1	0
0	1	1	0	1	0	1
0	1	0	0	1	0	1
0	0	1	0	1	0	1
0	0	0	0	1	0	1

CONTINGENT, SATISFIABLE, NOT A LOGICAL TRUTH

Next>>

p	q	r	p → q	q → r	(p → q) ∧ (q → r)	p → r	(p → q) ∧ (q → r) → (p → r)
1	1	1	1	1	1	1	1
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	1
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	1
0	1	0	1	0	0	1	1
0	0	1	1	1	1	1	1
0	0	0	1	1	1	1	1

p	q	r	p → q	q → r	p → q ∧ q → r	p → r	¬(p → r)
1	1	1	1	1	1	1	0
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	0
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	0
0	1	0	1	0	0	1	0
0	0	1	1	1	1	1	0
0	0	0	1	1	1	1	0

p	q	r	¬r	¬q	p ∨ q	¬(p ∨ q)	¬r ∨ ¬q	¬(p ∨ q) ↔ ¬r ∨ ¬q
1	1	1	0	0	1	0	0	1
1	1	0	1	0	1	0	1	0
1	0	1	0	1	1	0	1	0
1	0	0	1	1	1	0	1	0
0	1	1	0	0	1	0	0	1
0	1	0	1	0	1	0	1	0
0	0	1	0	1	0	1	1	1
0	0	0	1	1	0	1	1	1

p	q	r	p → q	q → r	(p → q) ∧ (q → r)	p → r	(p → q) ∧ (q → r) → (p → r)
1	1	1	1	1	1	1	1
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	1
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	1
0	1	0	1	0	0	1	1
0	0	1	1	1	1	1	1
0	0	0	1	1	1	1	1

p	q	r	p → q	q → r	p → q ∧ q → r	p → r	¬(p → r)
1	1	1	1	1	1	1	0
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	0
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	0
0	1	0	1	0	0	1	0
0	0	1	1	1	1	1	0
0	0	0	1	1	1	1	0

p	q	r	¬r	¬q	p ∨ q	¬(p ∨ q)	¬r ∨ ¬q	¬(p ∨ q) ↔ ¬r ∨ ¬q
1	1	1	0	0	1	0	0	1
1	1	0	1	0	1	0	1	0
1	0	1	0	1	1	0	1	0
1	0	0	1	1	1	0	1	0
0	1	1	0	0	1	0	0	1
0	1	0	1	0	1	0	1	0
0	0	1	0	1	0	1	1	1
0	0	0	1	1	0	1	1	1

Using the truth table method, determine whether these formulas are (1) tautological, contingent, or contradictory (2) Logical truth or not (3) Satisfiable or not:

Using the truth table method, determine whether
these formulas are (1) tautological, contingent, or contradictory (2) Logical truth or not (3) Satisfiable or not:

p	q	r	p → q	q → r	(p → q) ∧ (q → r)	p → r	(p → q) ∧ (q → r) → (p → r)
1	1	1	1	1	1	1	1
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	1
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	1
0	1	0	1	0	0	1	1
0	0	1	1	1	1	1	1
0	0	0	1	1	1	1	1

p	q	r	p → q	q → r	p → q ∧ q → r	p → r	¬(p → r)
1	1	1	1	1	1	1	0
1	1	0	1	0	0	0	1
1	0	1	0	1	0	1	0
1	0	0	0	1	0	0	1
0	1	1	1	1	1	1	0
0	1	0	1	0	0	1	0
0	0	1	1	1	1	1	0
0	0	0	1	1	1	1	0

p	q	r	¬r	¬q	p ∨ q	¬(p ∨ q)	¬r ∨ ¬q	¬(p ∨ q) ↔ ¬r ∨ ¬q
1	1	1	0	0	1	0	0	1
1	1	0	1	0	1	0	1	0
1	0	1	0	1	1	0	1	0
1	0	0	1	1	1	0	1	0
0	1	1	0	0	1	0	0	1
0	1	0	1	0	1	0	1	0
0	0	1	0	1	0	1	1	1
0	0	0	1	1	0	1	1	1