The domain of a relation R is the set of integers. x is related to y under relation R if x2=y. Select the description that accurately describes relation R.

Question

BlackTom AI · Accepted Answer

For integers, xRy exactly when x^2 = y.

1. Relevant concepts
- Reflexive: for every x, xRx (i.e., x^2 = x for all integers x).
- Anti-reflexive (irreflexive): for every x, not xRx (no integer satisfies x^2 = x).
- Symmetric: if xRy then yRx (i.e., x^2 = y implies y^2 = x).
- Anti-symmetric: if xRy and yRx then x = y (no distinct integers a, b with both a^2 = b and b^2 = a).

2. Step-by-step reasoning and intermediate calculations
- Check reflexive: solve x^2 = x => x^2 − x = 0 => x(x − 1) = 0 so x = 0 or x = 1. Since not every integer satisfies this, R is not reflexive.
- Check anti-reflexive: because 0R0 and 1R1 hold (0^2 = 0 and 1^2 = 1), R is not anti-reflexive.
- Check symmetric: symmetry would require that whenever x^2 = y, also y^2 = x. Counterexample: take x = 2, then 2R4 holds (2^2 = 4) but 4R2 does not (4^2 = 16 ≠ 2). Thus R is not symmetric.
- Check anti-symmetric: assume aRb and bRa, so a^2 = b and b^2 = a. Substitute b = a^2 into b^2 = a to get (a^2)^2 = a → a^4 = a → a^4 − a = 0 → a(a^3 − 1) = 0. Over integers this gives a = 0 or a^3 = 1 → a = 1. For a = 0, b = a^2 = 0; for a = 1, b = 1. Thus any pair with aRb and bRa must have a = b. There are no distinct integers a ≠ b satisfying both relations.

3. Why the correct answer is correct
Because the only solutions to the simultaneous conditions a^2 = b and b^2 = a are a = b, the formal definition of anti-symmetry is satisfied for all integers.

4. Why other options are incorrect
- Reflexive: false, since most integers (e.g., 2) do not satisfy x^2 = x.
- Anti-reflexive: false, since 0R0 and 1R1 hold.
- Symmetric: false, e.g., 2R4 but not 4R2.

Conclusion: Relation R is anti-symmetric, which matches the provided answer "Anti-symmetric."

The domain of a relation R is the set of integers. x is related to y under relation R if x2=y. Select the description that accurately describes relation R.单项选择题

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