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 one element.
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 one element.
The correct option is A 23
Total number of functions from A to B is 34 = 81. The number of functions which do not contain x or y or z in its range is 24. Therefore, the number of functions which contain exactly 2 elements in the range is 3.24 = 48.
The number of functions which contain exactly one element in its range is 3.Hence, the total number of onto functions is 81 – 48 + 3 = 36 => n(F) = 36.
Let f ϵ F. We now count the no. of ways in which f−1 consists of singe element. We can choose pre image of x in 4 ways. The remaining 3 elements can be mapped onto {y, z} is 23 – 2 = 6 ways. f−1 will consists of exactly one element in 4 × 6 = 24 ways. Thus, the probability of the require event is 2436=23.
Total number of functions from A to B is 34 = 81. The number of functions which do not contain x or y or z in its range is 24. Therefore, the number of functions which contain exactly 2 elements in the range is 3.24 = 48.
The number of functions which contain exactly one element in its range is 3.Hence, the total number of onto functions is 81 – 48 + 3 = 36 => n(F) = 36.
Let f ϵ F. We now count the no. of ways in which f−1 consists of singe element. We can choose pre image of x in 4 ways. The remaining 3 elements can be mapped onto {y, z} is 23 – 2 = 6 ways. f−1 will consists of exactly one element in 4 × 6 = 24 ways. Thus, the probability of the require event is 2436=23.
