GATE | GATE-CS-2015 (Set 2) | Question 65

The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is ________________
(A) 36
(B) 64
(C) 81
(D) 72


Answer: (A)

Explanation: A function f from X to Y is called onto if for all ‘y’ in Y there is an ‘x’ in X such that f(x) = y.

        
In onto functions, all elements in Y are used.

Source: http://www.regentsprep.org/regents/math/algtrig/atp5/OntoFunctions.htm

Every Surjective or Onto function sends two elements of {1, 2, 3, 4} to the same element of {a, b, c}. There are 4C2 = 6 such pairs of elements. The pairs are {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.

For a given pair {i, j} ⊂ {1, 2, 3, 4}, there are 3! sujective functions such that f(i) = f(j). Hence there are total 6 * 6 = 36 surjective functions.

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