Let S={1, 2, 3,4}. Then the number of elements in the set {f: S x S 
S : f is onto and f(a, b) = f(b, a) 
} is
Text Solution
Verified by Experts37
(37)
(1, 1), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4) - all have one choice for image.
(2, 1), (1, 2), (2, 2) - all have three choices for image
(3, 2), (2, 3), (3, 1), (1, 3), (3, 3) - all have two choices for image.
So the total functions = 3x3x2x2x2 = 72
Case 1 : None of the pre-images have 3 as image
Total functions = 2x2x1x1x1 = 4
Case 2 : None of the pre-images have 2 as image
Total functions = 2x2x2x2x2 = 32
Case 3 : None of the pre-images have either 3 or 2 as image
Total functions =1x1x1x1x1 = 1
Total onto functions = 72 - 4 - 32 + 1 = 37
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