Let N denote the set of all positive integers. Consider the sets A = \ 1, 2, 3, 4, 5\ and B = \ 1, 2, 3, 4, 5, 6, 7\ . Let S be the set of all functions f: A B such that f(2) 2 and f(4) 4. Consider the set T = \ f S : there exists a function g : B N such that g(f(x)) = 2^x for all x A\ . Then the number of elements in the set T is ___________.
A. A
The condition g(f(x)) = 2^x for all x A implies that if f(x_1) = f(x_2), then g(f(x_1)) = g(f(x_2)), which gives 2^ x_1 = 2^ x_2 x_1 = x_2. Thus, f must be an injective (one-to-one) function. Conversely, if f is injective, such a function g can always be constructed. Therefore, T is the set of all injective functions f: A B such that f(2) 2 and f(4) 4. The total number of injective functions from A to B is ^ 7 P_ 5 = 7 6 5 4 3 = 2520. Let E_2 be the set of injective functions where f(2) = 2. The number of such functions is ^ 6 P_ 4 = 6 5 4 3 = 360. Let E_4 be the set of injective functions where f(4) = 4. The number of such functions is ^ 6 P_ 4 = 360. Let E_2 E_4 be the set of injective functions where both f(2) = 2 and f(4) = 4. The number of such functions is ^ 5 P_ 3 = 5 4 3 = 60. Using the Principle of Inclusion-Exclusion, the number of injective functions where f(2) = 2 or f(4) = 4 is: |E_2 E_4| = |E_2| + |E_4| - |E_2 E_4| = 360 + 360 - 60 = 660 The number of elements in the set T is the number of injective functions where f(2) 2 and f(4) 4, which is: |T| = 2520 - 660 = 1860 Answer: 1860
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