ITPEC FE Morning April 2020 Question 23

XOR Boolean Expression — identify the correct algebraic form of XOR.

XOR means "one or the other, but not both."

Canonical form:
A ⊕ B = (A · B̄) + (Ā · B)

Equivalent product-of-sums form:
A ⊕ B = (A + B) · NOT(A · B)

This reads as: OR (at least one is true) AND NAND (not both are true).

Derivation:
- NOT(A · B) = NAND = "not both true"
- (A + B) = OR = "at least one true"
- Together: "at least one, but not both" = XOR

Why not others:
- (a) (A · B) · NOT(A · B) — AND · NAND = always 0 (contradicts itself)
- (c) (A · B) · NOT(A + B) — AND · NOR = always 0 (AND needs at least one true, NOR needs both false)
- (d) (A + B) · NOT(A + B) — OR · NOR = always 0 (contradicts itself)

Key rule: XOR = OR · NAND = (A + B) · NOT(A · B)

AI-generated — may contain errors