# Question: let px and qx be logical statements depending on the...

###### Question details

Let P(x) and Q(x) be logical statements depending on the variable x in the nonempty set S. For each of the following pairs of logical statements D and E, give a proof or counterexample for the statements D ⇒ E and E ⇒ D. ( all that is explained in the question)

(a) ∀(x ∈ S)[P(x) ⇔ Q(x)] and [∀(x ∈ S)P(x)] ⇔ [∀(x ∈ S)Q(x)]

(b) ∃(x ∈ S)[P(x) ⇔ Q(x)] and [∃(x ∈ S)P(x)] ⇔ [∃(x ∈ S)Q(x)]

(c) ∀(x ∈ S)[P(x)∧Q(x)] and [∀(x ∈ S)P(x)]∧[∀(x ∈ S)Q(x)]

(d) ∀(x ∈ S)[P(x)∨Q(x)] and [∀(x ∈ S)P(x)]∨[∀(x ∈ S)Q(x)]