The relations ≤ and < on a Boolean algebra are defined as: x ≤ y if and only if x ∨ y = y; x < y means x ≤ y but x ≠ y; x ≥ y means y ≤ x; and x > y means y < x. Considering these definitions, which of the following is not true in the Boolean algebra?

(i) If x ≤ y and y ≤ z, then x ≤ z

(ii) If x ≤ y and y ≤ x, then x = y

(iii) If x < y and y < z, then x ≤ y

(iv) If x < y and y < z, then x < y

1. A(iii) only
2. B(iv) only
3. C(i) and (ii) only
4. D(ii) and (iii) only ¯A¯¯¯ ⋅ B + A ⋅ ¯B¯¯¯ + A ⋅ B

Pick an option to check yourself, or open the answer below.

Show answer & explanation

Correct answer: A. (iii) only

Explanation

≤ defined by x ∨ y = y is a partial order: it is transitive (i) and antisymmetric (ii). In (iii) and (iv) the conclusion is already one of the hypotheses (x < y implies x ≤ y), so both hold too.

Note: no option is fully correct; the official key (A) is shown. As printed, all four statements are true (≤ is a partial order, and (iii)/(iv) follow directly from x < y). No option is literally right; the official key is (A) (iii) only.