Let A={a,b}, B={c,d,e}. The total number of possible relations from A to B is?

Question

Let A={a,b}, B={c,d,e}. The total number of possible relations from A to B is?

BlackTom AI · Accepted Answer

A relation from A to B is any subset of the Cartesian product A×B, so count the ordered pairs in A×B and then count all subsets.

1) Relevant concepts:
- Cartesian product A×B = set of all ordered pairs (x,y) with x in A and y in B.
- A relation from A to B is any subset of A×B.
- The number of subsets of a finite set with k elements is 2^k (each element can be either included or excluded).

2) Step-by-step reasoning:
- |A| = 2 and |B| = 3.
- Number of ordered pairs in A×B = |A| × |B| = 2 × 3 = 6.
- List them explicitly: A×B = {(a,c), (a,d), (a,e), (b,c), (b,d), (b,e)}.

3) Calculations (intermediate steps shown):
- k = number of elements in A×B = 6.
- Number of possible relations = number of subsets of A×B = 2^k = 2^6.
- Compute 2^6 = 64.

4) Why this is correct:
Each of the 6 ordered pairs can independently either belong to a relation or not. That gives 2 choices per pair and thus 2×2×...×2 (six times) = 2^6 possible distinct relations, including the empty relation and the full relation (A×B).

5) Why other options are incorrect:
- 6: this counts the number of ordered pairs in A×B, not the number of possible subsets/relations.
- 15: equals 2^4 − 1 or similar small-set counts but has no relation to 2^6; it does not represent the number of subsets of a 6-element set.
- 5: unrelated to the problem (it equals |B|, but the number of relations depends on |A|×|B|).

6) Conclusion:
Therefore the total number of possible relations from A to B is 64, which matches the provided correct answer.

Let A={a,b}, B={c,d,e}. The total number of possible relations from A to B is?单项选择题

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