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Tautologies

A tautology is a statement that is always true.




Examples:




We are always looking for tautologies in math. As shown in the second example, we could always replace the statement "P=Q" with the statement "PQ is a tautology".

In lesson 1.5, we stated the transitive property as ((PQ)(QR))(PR) is always true. Instead, we could have said it is a tautology.

Contradictions

A contradiction is a statement that is always false.




Examples:




We use F to represent a contradiction. It is not the same as F, because F is a value in the T,F space whereas F is a statement that is false no matter what values you plug in.

The negation of a contradiction, ¬F, is a tautology. So, finding a contradiction is essentially equivalent to finding a tautology since we can always negate the contradiction. The first example of a tautology above is the negation of the first example of a contradiction: ¬(P¬P)=P¬P

Examples

The statement T is a tautology, and the statement F is a contradiction.

Claim: PT=¬P

We show the claim is true with a truth table:

P
T
F
T
T
T
¬P
F
T
PT
F
T
Notice that there are only 2 rows, even though there are 2 variables. This is because T can only be true, so all possible inputs for P and T can be written in 2 rows.

Also, let's replace the = with .
¬P
F
T
PT
F
T
¬PPT
T
T
So, since PT=¬P, PT¬P is a tautology.




Claim: FP=T

We show the claim is true with a truth table:

P
T
F
F
F
F
T
T
T
FP
T
T

Try entering values for P.

PT= T

PF= T

TP= T

PF= T

PF= T