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Composite Function

Let A and B b...

Question

Let $A$ and $B$ be two sets such that $n (A) = 6$ and $n (B) = 3,$ then number of onto functions from $A$ to $B$ is

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Solution

If $n (A) = m$ and $n (B) = n,$ where $1 \leq n \leq m,$ then number of onto functions from $A$ to $B$
$= n \sum r = 1 (- 1)^{n - r}^{n} C_{r} \cdot r^{m}$

Here, $m = 6, n = 3$
$∴$ Number of onto functions
$= 3 \sum r = 1 (- 1)^{3 - r}^{3} C_{r} \cdot r^{6}$
$=^{3} C_{1} -^{3} C_{2} (2^{6}) +^{3} C_{3} (3^{6})$
$= 3 - 192 + 729$
$= 540$

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Similar questions

Let A and B be two sets such that n(A) = p and n(B) = q, write the number of functions from A to B.

Let A and B be two non empty sets such that $n (A) = 5, n (B) = 6$ and $n (A \cap B) = 3$ .

Find (i) $n (A \times B)$ ,

(ii) $n (B \times A)$ and

(iii) $n (A \times B) \cap (B \times A)$

A

B

n (A) = 70, n (B) = 60

n (A \cup B) = 110

n (A \cap B)

A

B

n (A) = 6

n (B) = 3

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