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JEE Main202127 Jul 2021Morning ShiftMathematicsFunctionsActual

Let S = 1 , 2 , 3 , 4 , 5 , 6 , 7 . Then the number of possible functions f : S → S such that f m · n = f m · f n for every m , n ∈ S and m · n ∈ S , is equal to _____.

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Step-by-step solution

Given, f m n = f m · f n Put m = 1 , then f 1 · n = f 1 · f n ⇒ f 1 = 1   f n ≠ 0 Put m = n = 2 f 4 = f 2 · f 2 If f 2 = 1 , then f 4 = 1 and if f 2 = 2 , then f 4 = 4 Put m = 2 , n = 3 f 6 = f 2 · f 3 If f 2 = 1 , then f 3 can be any value between 1 to 7 If f 2 = 2 , then f 3 can be 1 or 2 or 3 Similarly f 5 , f 7 can take any value from 1 , 2 , . . . , 7 So, total number of function = 1 × 1 × 7 × 1 × 7 × 1 × 7 + 1 × 1 × 3 × 1 &

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