Propositional Logic
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 ).
Hence
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