Propositional Logic

Examples of Use of Identities

Here a few examples are presented to show how the identities in Identities can be used to prove some useful results.

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 Proof of Implications.

Next -- Implications

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