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Let S = 1 , 2 , 3 , 4 . Then the number of elements in the set f : S × S → S : f is onto and f a , b = f b , a ≥ a ∀ a , b ∈ S × S is

Correct answer

37

Step-by-step solution

Finding the cases for f a , b = f b , a ≥ a we have, 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 × 3 × 2 × 2 × 2 = 72 Case 1: None of the pre-images have 3 as image. Total functions = 2 × 2 × 1 × 1 × 1 = 4 Case 2: None of the pre-images have 2 as image Total functio

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