## Content

### Tautologies

There are certain truth tables where all the entries are true (1). We found such an example in Q51(v): the proposition $(q \to p) + (p \to q) $ is always true. Propositions that are always true or valid are called tautologies.

It is through tautologies that we can decide whether or not an argument is valid. If the truth table for the argument shows that the overall proposition is a tautology, then the argument is valid. Suppose that $p$ is the statement 'it is hot'. Then $p + p'$ tells us that either it is hot or it is not hot'. As Q54(i) is a tautology, then, as we would expect, either it is hot or it's not to be one too. In discussions or even written statements, things often get complicated so quickly that it is hard to see if the overall argument is valid. The truth tables of such arguments allow us to decide if they are true or not by seeing whether or not they are tautologies.

Questions

- Decide which of the following arguments are valid:
- $\hspace{3px}$When it rains the roof gets wet. The roof is wet. Therefore it has rained;
- $\hspace{2px}$In a right angled triangle with sides $a, b$ and $c$ with $c$ the hypotenuse, $a^2 + b^2 = c^2$. Therefore the triangle with sides 3, 4 and 5 is a right angled triangle;
- In a right angled triangle with sides $a, b$ and $c$ with $c$ the hypotenuse, $a^2 + b^2 = c^2$. Therefore the triangle with sides 3, 4 and 6 is not a right angled triangle.