Question

For each of the following, use logic laws to decide whether the statement is a tautology contradiction, or neither. a (a) (A B) A (B A) (b) (PP) AP

Answer 1

The statement
`(A\rightarrow \lnot B)\land (B\rightarrow A)` is a contingency.

The statement
`(P\rightarrow \lnot P)\land P` is a contradiction.

Explanation:

A tautology is a proposition that is always true.

A contradiction is a proposition that is always false.

A contingency is a proposition that is neither a tautology nor a contradiction.

a) To classify the statement
`(A\rightarrow \lnot B)\land (B\rightarrow A)`, you need to use the logic laws as follows:

`(A\rightarrow \lnot B)\land (B\rightarrow A) \equiv`

`\equiv (\lnot A \lor\lnot B)\land(\lnot B \lor A)` by the logical equivalence involving conditional statement.

`\equiv (\lnot B\lor \lnot A )\land(\lnot B \lor A)` by the Commutative law.

`\equiv \lnot B \lor (\lnot A \land A)` by Distributive law.

`\equiv \lnot B \lor (A \land \lnot A)` by the Commutative law.

`\equiv \lnot B \lor F` by the Negation law.

Therefore the statement
`(A\rightarrow \lnot B)\land (B\rightarrow A)` is a contingency.

b) To classify the statement
`(P\rightarrow \lnot P)\land P`, you need to use the logic laws as follows:

`(P\rightarrow \lnot P)\land P \equiv`

`\equiv (\lnot P \lor \lnot P)\land P` by the logical equivalence involving conditional statement.

`\equiv P \land (\lnot P \lor \lnot P)` by the Commutative law.

`\equiv (P \land \lnot P) \lor (P \land \lnot P)` by Distributive law.

`\equiv F \lor F \equiv F` by the Negation law.

Therefore the statement
`(P\rightarrow \lnot P)\land P` is a contradiction.