Page 35 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Truth Tables 21
DeMorgan’s laws
¬(P ∧ Q) is equivalent to ¬P ∨¬Q.
¬(P ∨ Q) is equivalent to ¬P ∧¬Q.
Commutative laws
P ∧ Q is equivalent to Q ∧ P.
P ∨ Q is equivalent to Q ∨ P.
Associative laws
P ∧ (Q ∧ R) is equivalent to (P ∧ Q) ∧ R.
P ∨ (Q ∨ R) is equivalent to (P ∨ Q) ∨ R.
Idempotent laws
P ∧ P is equivalent to P.
P ∨ P is equivalent to P.

Distributive laws
P ∧ (Q ∨ R) is equivalent to (P ∧ Q) ∨ (P ∧ R).
P ∨ (Q ∧ R) is equivalent to (P ∨ Q) ∧ (P ∨ R).

Absorption laws
P ∨ (P ∧ Q) is equivalent to P.
P ∧ (P ∨ Q) is equivalent to P.
Double Negation law

¬¬P is equivalent to P.

Notice that because of the associative laws we can leave out parentheses in
formulas of the forms P ∧ Q ∧ R and P ∨ Q ∨ R without worrying that the
resulting formula will be ambiguous, because the two possible ways of filling
in the parentheses lead to equivalent formulas.
Many of the equivalences in the list should remind you of similar rules in-
volving +, ·, and − in algebra. As in algebra, these rules can be applied to more
complex formulas, and they can be combined to work out more complicated
equivalences. Any of the letters in these equivalences can be replaced by more
complicated formulas, and the resulting equivalence will still be true. For ex-
ample, by replacing P in the double negation law with the formula Q ∨¬R,
you can see that ¬¬(Q ∨¬R) is equivalent to Q ∨¬R. Also, if two formulas
are equivalent, you can always substitute one for the other in any expression
and the results will be equivalent. For example, since ¬¬P is equivalent to