Question

Use logical equivalences (not a truth table) to reduce p → (q − p) to a tautology t. In other words, you should transform p → (q − p) into an equivalent statement, then transform that into another equivalent statement, and so on, until you arrive at a tautology. Your solution should look something like this: p → (q − p) = statement = statement = . . . . ... = t. (Note: the symbol = is technically the same as H. It's just easier to use = , because can easily be confused as being part of the logical statement you're transforming).

Answer 1

The statement
`p\rightarrow (q\rightarrow p)` is a tautology.

`p\rightarrow (q\rightarrow p)\equiv \lnot p\lor (q\rightarrow p) \equiv \lnot p\lor (\lnot q\lor p) \equiv (p\lor \lnot p) \lor \lnot q \equiv T \lor \lnot q \equiv T`

Explanation:

We have the following statement:

`p\rightarrow (q\rightarrow p)`

To reduce the statement to a tautology we need to use the table of logical equivalences as follows:

`p\rightarrow (q\rightarrow p)\equiv`

`\equiv \lnot p\lor (q\rightarrow p)` by the the logical equivalence involving conditional statement.

`\equiv \lnot p\lor (\lnot q\lor p)` by the the logical equivalence involving conditional statement.

`\equiv (p\lor \lnot p) \lor \lnot q` by the Associative law.

`\equiv T \lor \lnot q` by the Negation law.

`\equiv T` by the Domination law.