Page 37 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/
0521861241c01 CB996/Velleman October 19, 2005 23:46 0 521 86124 1 Char Count= 0

Truth Tables 23
From the truth table it is clear that the first formula is a tautology, the second
a contradiction, and the third neither. In fact, since the last column is identical
to the first, the third formula is equivalent to P.
We can now state a few more useful laws involving tautologies and contradic-
tions.Youshouldbeabletoconvinceyourselfthatalloftheselawsarecorrectby
thinking about what the truth tables for the statements involved would look like.
Tautology laws
P ∧ (a tautology) is equivalent to P.
P ∨ (a tautology) is a tautology.
¬(a tautology) is a contradiction.

Contradiction laws
P ∧ (a contradiction) is a contradiction.
P ∨ (a contradiction) is equivalent to P.
¬(a contradiction) is a tautology.

Example 1.2.7. Find simpler formulas equivalent to these formulas:
1. P ∨ (Q ∧¬P).
2. ¬(P ∨ (Q ∧¬R)) ∧ Q.

Solutions
1. P ∨ (Q ∧¬P)
is equivalent to (P ∨ Q) ∧ (P ∨¬P) (distributive law),
which is equivalent to P ∨ Q (tautology law).
The last step uses the fact that P ∨¬P is a tautology.
2. ¬(P ∨ (Q ∧¬R)) ∧ Q
is equivalent to (¬P ∧¬(Q ∧¬R)) ∧ Q (DeMorgan’s law),
which is equivalent to (¬P ∧ (¬Q ∨¬¬R)) ∧ Q (DeMorgan’s law),
which is equivalent to (¬P ∧ (¬Q ∨ R)) ∧ Q (double negation law),
which is equivalent to ¬P ∧ ((¬Q ∨ R) ∧ Q) (associative law),
which is equivalent to ¬P ∧ (Q ∧ (¬Q ∨ R)) (commutative law),
which is equivalent to ¬P ∧ ((Q ∧¬Q) ∨ (Q ∧ R))
(distributive law),
which is equivalent to ¬P ∧ (Q ∧ R) (contradiction law).

The last step uses the fact that Q ∧¬Q is a contradiction. Finally, by the
associative law for ∧ we can remove the parentheses without making the
formula ambiguous, so the original formula is equivalent to the formula
¬P ∧ Q ∧ R.