Subsection 3.3.2 A List of Identities
In Boolean logic, we also have a set of identities that enable us to transform expressions without changing their values (in this case, their truth values). Some of them are analogs of the arithmetic identities. For example, both or and and are commutative. Some will be new.
Here’s a list of identities that we’ll find most useful. The way to prove them is to use truth tables to show that the truth values of both sides are the same. We’ll prove the first one. We suggest that you prove at least a few more of them. It will give you practice using truth tables and you’ll find yourself proving your first useful theorems.
Notice that each of these identities has the form:
expression 1 ≡ expression 2
Recall that the symbol ≡ means is equivalent to . What we’re claiming, in our statement of each of these identities, is that the two sides of the equivalence statement have the same truth values. And we’re claiming that this holds for all propositions ( p , q , r , or even ones that themselves contain Boolean operators).