Answer:
The statement [tex](A\rightarrow \lnot B)\land (B\rightarrow A)[/tex] is a contingency.
The statement [tex](P\rightarrow \lnot P)\land P[/tex] is a contradiction.
Step-by-step 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 [tex](A\rightarrow \lnot B)\land (B\rightarrow A)[/tex], you need to use the logic laws as follows:
[tex](A\rightarrow \lnot B)\land (B\rightarrow A) \equiv[/tex]
[tex]\equiv (\lnot A \lor\lnot B)\land(\lnot B \lor A)[/tex] by the logical equivalence involving conditional statement.
[tex]\equiv (\lnot B\lor \lnot A )\land(\lnot B \lor A)[/tex] by the Commutative law.
[tex]\equiv \lnot B \lor (\lnot A \land A)[/tex] by Distributive law.
[tex]\equiv \lnot B \lor (A \land \lnot A)[/tex] by the Commutative law.
[tex]\equiv \lnot B \lor F[/tex] by the Negation law.
Therefore the statement [tex](A\rightarrow \lnot B)\land (B\rightarrow A)[/tex] is a contingency.
b) To classify the statement [tex](P\rightarrow \lnot P)\land P[/tex], you need to use the logic laws as follows:
[tex](P\rightarrow \lnot P)\land P \equiv[/tex]
[tex]\equiv (\lnot P \lor \lnot P)\land P[/tex] by the logical equivalence involving conditional statement.
[tex]\equiv P \land (\lnot P \lor \lnot P)[/tex] by the Commutative law.
[tex]\equiv (P \land \lnot P) \lor (P \land \lnot P)[/tex] by Distributive law.
[tex]\equiv F \lor F \equiv F[/tex] by the Negation law.
Therefore the statement [tex](P\rightarrow \lnot P)\land P[/tex] is a contradiction.
