Truth Tables
Truth tables are useful for showing the complete behavior of a
Boolean function.
However, hand construction of a truth table is tedious once
there are more than a few propositions in the expression.
The number of truth assignments is exponential in the number
of propositions.
Truth Tables
You know that there are 2n truth assignments on n Boolean
variables. A (complete) truth table shows the input/output behavior
for all possible truth assignments. The rate of growth in a truth
table rows as a functions of the number of propositions is shown
in the table below.
Number of Number of
Propositions
Rows
0
1
2
3
4
..
.
1
2
4
8
16
..
.
n
2n
Truth Tables
There is no one way to construct a truth table.
However, basis rules exist. The rules are similar to those for
performing arithmetic operations.
1. Expressions in parenthesis are evaluated first from innermost
to outermost.
2. NOT is evaluated before AND
3. AND is evaluated before OR
4. OR is evaluated before IMPLIES
5. IMPLIES is evaluated before EQUIV
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Tautologies and Valid Statements
When a Boolean function is True for every one of its truth
assignments, the expression is said to be a “tautology.” Or the
statement is valid.
A valid statement is always True.
There are several important valid statements.
• p ∨ ¬p Exclusive Middle
• (p ∧ (p → q)) → q Modus Ponens
• p ∧ (¬q → ¬p) → q Modus Tollens
• ((¬p → q) ∧ (¬p → ¬q)) → p Reductio ad Absurdum
• ((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r) Resolution
Reductio Ad Absurdum Truth Table
Let’s construct a truth table for Reductio ad absurdum.
((¬p → q) ∧ (¬p → ¬q)) → p
p
Input
q
0
0
1
1
0
1
0
1
Reductio Ad Absurdum
((¬p → q) ∧ (¬p → ¬q)) →
p
0
1
1
1
0
0
1
1
1
0
1
1
1
1
1
1
0
0
1
1
7
1st
3
3rd
11
2nd
15
5th
3
4th
2
Resolution Truth Table
Let’s construct a truth table for resolution.
((p ∨ q) ∧ (¬p ∨ r)) → (q ∨ r)
p
Input
q r
((p ∨ q)
∧
Resolution
(¬p ∨ r)) →
(q ∨ r)
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
1
1
1
1
0
0
1
1
0
1
0
1
1
1
1
1
0
1
0
1
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
1
15
51
85
63
1st
53
3rd
245
2nd
255
5th
119
4th
Contradiction and Invalid Statements
When a Boolean function is False for every one of its truth
assignments, the expression is said to be a contradiction.
Of course, you can construct a contradiction by negating a tautology.
A computational hurdle occurs when every truth assignment must
be investigated to determine if a statement is a tautology or a
contradiction.
Contingency Statements
When a Boolean function sometimes True and sometimes False,
dependent on the values in a truth assignment, the expression is said
to be a contingency.
Satisfiability is a fundamental problem in computer science.
Problem 1 (SAT). Give a Boolean function in n variables is it satisfiable?
That is, is there a truth assignment for its propositions that makes the expression
True?
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Tautology, Contradictions or Contingency?
Consider the Boolean function
B(p, q, r) = ((p ∨ q) ∧ r) → p
Is it a tautology? (always True).
Is it a contradiction? (always False).
Or, is it a contingency? (sometimes True and sometimes False).
Let’s construct a truth table to find out.
Truth Table for a Boolean Expression
A truth table for ((p ∨ q) ∧ r) → p is:
p
0
0
0
0
1
1
1
1
Input
q r
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
((p ∨ q)
Expression
∧
r
→
p
0
0
1
1
1
1
1
1
0
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1st
3rd
2nd
5th
4th
The expression (function) is a contingency.
4