# MandysNotes

Sunday, 06 April 2014 00:00

## Tautology

By  Gideon
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Imagine your friend tells you that the animal he is thinking of is either duck-billed or not duck-billed.

In other words, he is telling you that either $$A$$ is true, or $$\lnot A$$ is true.

Equivalently he is telling you that the statement:

$A \cup \lnot A$

is true.

This is not very helpful in geussing which animal he is thinking of, but it does have the property of always being true.

In other words, the statement

$A \cup \lnot A$

is true if $$A$$ is true (and hence $$\lnot A$$ is false);

it is also true if $$A$$ is false (and hence $$\lnot A$$ is true).

Such a statement is called a tautology.

A tautology must have a truth table with all "T"s under it.

The truth table for $$A \cup \lnot A$$ is as follows: