In order to describe a chance experiment, we have used the following (continuous)

sample space

Ω = {(x, y) ∶ −5 ≤ x ≤ 5 and − 3 ≤ y ≤ 7}.

On this space, we define the following events:

A = {(x, y) ∈ Ω ∶ x = y}, B = {(x, y) ∈ Ω ∶ x2 = y2}, C = {(x, y) ∈ Ω ∶ x + y = 0}, D = {(x, y) ∈ Ω ∶ x ≤ y}, E = {(x, y) ∈ Ω ∶ x ≥ y}.

Which of the following statements are correct and which are false?

(a) B = AC;

(b) AC ⊆ B;

(c) D and E are mutually exclusive events;

(d) A = {(x, x) ∶ |x| ≤ 5};

(e) A = DE;

(f) D′ = E − A;

(g) A = {(x, x) ∶ −3 ≤ x ≤ 3}.