Show that PPQ is a tautology What is the connection between

Show that [P¬P]Q is a tautology. What is the connection between this statement being a tautology and the truth table for the conditional (if, then statements)?

Solution

So, we have 2 variables P and Q, and the relation is called a tautology if all its Values are True.

The Truth Table for AND is

p

q

p^q

0

0

0

0

1

0

1

0

0

1

1

1

Truth Table for pq is

p

q

pq

F

F

T

F

T

T

T

F

F

T

T

T

The Desired Truth table is: -

p

q

      ¬p

p^¬p

(p^¬p )q

F

F

T

F

T

F

T

T

F

T

T

F

F

F

T

T

T

F

F

T

So as all the values of (p^¬p )q are True , it is a Tautology.

The Truth Table for the conditional statement is False only for a specific condition, when 1st variable is true and the 2nd is false.

Hence prooved the given Variable is a tautology.

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p	q	p^q
0	0	0
0	1	0
1	0	0
1	1	1