Prove Logical Operations Using Inference Rules

Question

Premise 1: p ∧ q

Premise 2: q → r

Premise 3: s → ¬r

Premise 4: ¬r → ¬u

Premise 5: t ∨ s

Premise 6: t → ¬p ∨ U

Prove: u ∧ q

Does anybody know how to solve this proof using rules of inference? I know the rules of inference like modus ponens/tollens but I am not sure how to use them here. I am still beginning to learn these types of proofs.

Can anybody show me how to complete this? Thanks.

Answer 1

Since p ∧ q → p and p ∧ q → q, by Premise 1 both p and q are true.

By Premise 2 we now know that r is true.

By Premise 3, r → ¬s, so s is false.

Then, by Premise 5, t must be true.

Now, by Premise 6, ¬p ∨ u is true, but since p is true, it is u which must be true.

Finally, both q and u are true and so it is u ∧ q.

(Also note that Premise 4 is not needed)