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 ).**

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
**
Proof of Implications**.

**
Next -- Implications
**

Back to Schedule

Back to Table of Contents