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Question: Let F denote the set of all onto functions from A = {a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub>, a...

Let F denote the set of all onto functions from

A = {a1, a2, a3, a4} to B = {x, y, z}. A function f is chosen at random from F. The probability that f –1 (x) consists of exactly two elements is

A

2/3

B

1/3

C

1/6

D

0

Answer

2/3

Explanation

Solution

Let us first count the number of elements in F. Total number of functions from A to B is 34 = 81.

The number of functions which do not contain

x(y) [z] in its range is 24.

\the number of functions which contain exactly two

elements in the range is 3 . 24 = 48.

The number of functions which contain exactly one element in its range is 3.

Thus, the number of onto functions from A to B is 81 – 48 + 3 = 36 [using principle of inclusion exclusion]

n (F) = 36.

Let f Ī F. We now count the number of ways in which f –1(x) consists of single element.

We can choose preimage of x in 4 ways. The remaining 3 elements can be mapped onto {y, z} is 23 – 2 = 6 ways.

\ f –1 (x) will consists of exactly one element in

4 × 6 = 24 ways.

Thus, the probability of the required event is 24/36 = 2/3