A binary relation R on a single set A is a subset of?

Question

BlackTom AI · Accepted Answer

A binary relation R on A is a set of ordered pairs (a, b) with a in A and b in A, so R ⊆ A x A.

1. Relevant concepts
- Ordered pair: (a, b) denotes the first element a and second element b.
- Cartesian product A x A: the set {(x, y) | x ∈ A and y ∈ A}. It contains every possible ordered pair whose components are both from A.
- Binary relation on A: any subset of A x A (i.e., any collection of ordered pairs drawn from A x A).

2. Step-by-step reasoning
- By definition, a binary relation on A relates two elements of A; each relation element is an ordered pair (a, b) with a ∈ A and b ∈ A.
- The collection of all such possible pairs is exactly A x A. A particular relation R picks some of those pairs, so R is a subset of A x A.
- Example: if A = {1, 2}, then A x A = {(1,1), (1,2), (2,1), (2,2)}. A relation R might be {(1,2), (2,2)}, clearly R ⊆ A x A.

3. Intermediate “calculation” steps (illustrative)
- List A = {1,2}
- Compute A x A = {(1,1), (1,2), (2,1), (2,2)}
- Choose R = {(1,2), (2,2)} → verify every element of R is in A x A.

4. Why the correct answer is correct
- Because every element of a binary relation on A is an ordered pair with both entries from A, the natural ambient set containing all such pairs is the Cartesian product A x A; hence R ⊆ A x A.

5. Why other options are incorrect
- "A - B": set difference between A and some B (not even defined here) is a subset of A, not a set of ordered pairs.
- "A + A": ambiguous notation (sometimes used informally for multiset sum or algebraic sum) and does not denote the Cartesian product of ordered pairs.
- "A → B": denotes a function type or arrow from A to B (mapping), not the set of ordered pairs within A itself; also it references two sets A and B, not the Cartesian product A x A.

6. Conclusion
The correct choice is A x A, which matches the provided answer.

A binary relation R on a single set A is a subset of?单项选择题

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