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Which of the following relations defined on the set of integers Z is NOT a partial order relation?

1. (A) R1 = {(a, b): a, b ∈ Z, a < b}
2. (B) R2 = {(a, b): a, b ∈ Z, a ≤ b}
3. (C) R3 = {(a, b): a, b ∈ Z, a = b}
4. (D) None of the above

Correct answer: (A) R1 = {(a, b): a, b ∈ Z, a < b}

Solution

The relation a < b is not reflexive (a < a is false), so R1 fails to be a partial order, whereas '<=' and '=' are both partial orders.

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