Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent, without using truth tables.

4. Show that (p → q) ∧ (p → r) and p → (q ∧ r) are logically equivalent using equivalencelaw.

Well-formed-formula (wff) Prove (p a q) and (T→ (¬ (p → q))) are both wff using top-down parsing:

please naswer

Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent, without using truth tables.

PLEASE TYP URGENT***

Please help me with this. I am having trouble understanding how to do this

Prove the language L = {1n | m ∈ N ∧n = 2m} is non-regular. Explain each step of your reasoning with precision.

Prove using the concept of interpretations and the value of a formula vI under an interpretation I that
[∀x p(x) ∨ ∃x q(x)] → ∃x[p(x) ∨ q(x)] is valid.

6. Use rules of inference to show that if Vr(P(x) v Q(x)) and Vr((-P(x) ^ Q(x)) → R(x)) are true, then Vx(-R(x) → P(x)) is also true, where the domains of all quantifiers are the same. 7. (a) Prove that m³ + 2n² = 36 has no solution in positive integers. (b) Prove that for every n e Z, n³ +n is even. (c) Prove that for all n e Z, n is odd if and only if n + 2 is odd. (d) Prove that the product of two consecutive integers is even.