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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) ≥ a ∀ (a,b) ∈ S x S} is |
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Answer» Correct answer is 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 = 3 x 3 x 2 x 2 x 2 = 72 Case 1 : None of the pre-images have 3 as image Total functions = 2 x 2 x 1 x 1 x 1 = 4 Case 2 : None of the pre-images have 2 as image Total functions = 2 x 2 x 2 x 2 x 2 = 32 Case 3 : None of the pre-images have either 3 or 2 as image Total functions = 1 x 1 x 1 x 1 x 1 = 1 ∴ Total onto functions = 72 – 4 – 32 + 1 = 37 |
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