Here a few examples are presented to show how the identities in
1. ( P Q ) ( P Q )
What this means is that the negation of "if P then Q" is "P but not Q". For example, if you said to someone "If I win a lottery, I will give you $100,000." and later that person says "You lied to me." Then what that person means is that you won the lottery but you did not give that person $100,000 you promised.
To prove this, first let us get rid of using one of the identities: ( P Q ) ( P Q ).
That is, ( P Q ) ( P Q ).
Then by De Morgan, it is equivalent to P Q , which is equivalent to P Q, since the double negation of a proposition is equivalent to the original proposition as seen in the identities.
2. P ( P Q ) P --- Absorption
What this tells us is that P ( P Q ) can be simplified to P, or if necessary P can be expanded into P ( P Q ) .
To prove this, first note that P ( P T ).
P ( P Q )
( P T ) ( P Q )
P ( T Q ) , by the distributive law.
( P T ) , since ( T Q ) T.
P , since ( P T ) P.
Note that by the duality
P ( P Q ) P also holds.
More examples of use of identities can be found in the