Let A be a set with 3 elements and B a set with 4 elements. The number of onto functions from A to B is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 0
A function from A to B is onto when every element of B is the image of some element of A.
A has only 3 elements, so any function from A to B has at most 3 different images.
B has 4 elements, so at least one element of B is always left without a pre-image.
Hence no function from A to B is onto, and the number of onto functions is 0.
Why the other options are incorrect
Option (1) 64 — Incorrect. This is \(4^{3}\), the number of all functions from A to B, none of which is onto.
Option (2) 24 — Incorrect. This is \(4 \times 3 \times 2\), the number of one-one functions from A to B, which are still not onto.
Option (4) 81 — Incorrect. This is \(3^{4}\), the number of functions from B to A, not from A to B.