Some of the rules of inference can be proven using the other rules of inference and the laws of propositional logic.

(a)One of the rules of inference is Modus tollens:

p q q p

Prove that Modus tollens is valid using the laws of propositional logic and any of the other rules of inference besides Modus tollens. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(b)One of the rules of inference is Modus ponens:

p q p q

Prove that Modus ponens is valid using the laws of propositional logic and any of the other rules of inference besides Modus ponens. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(c)One of the rules of inference is Disjunctive syllogism :

p q p q

Prove that Disjunctive syllogism is valid using the laws of propositional logic and any of the other rules of inference besides Disjunctive syllogism. (Hint: you will need one of the conditional identities from the laws of propositional logic).

(d)One of the rules of inference is Resolution:

p q p r q r

Prove that Resolution is valid using the laws of propositional logic and any of the other rules of inference besides Resolution. (Hint: you will need one of the conditional identities from the laws of propositional logic).

Please answer the proof via expression