ITPEC FE Morning October 2018 Question 1
ITPEC FE Morning October 2018 — Question 1 of 80
Exclusive OR (XOR) logical equivalence — expressing XOR using basic AND, OR, NOT operators.
XOR means "one or the other, but not both." Its canonical form using basic operators is:
p ⊕ q = (p ∨ q) ∧ ¬(p ∧ q)
This reads: "p or q is true" AND "it is not the case that both are true."
Verification via truth table:
| p | q | p ⊕ q | (p ∨ q) ∧ ¬(p ∧ q) |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
Why not others:
- (a) ¬p ∨ q is the material implication (p → q), not XOR. It yields 1 when p=0, q=0.
- (c) ¬(p ∨ q) ∨ (p ∧ q) is true when both inputs match (XNOR), the exact opposite of XOR.
- (d) (p ∧ ¬q) ∨ ¬(q ∧ p) evaluates to 1 when p=0, q=0 (since ¬(0 ∧ 0) = 1), but XOR gives 0.
Key rule: XOR = OR minus the "both true" case → (p ∨ q) ∧ ¬(p ∧ q).
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