Exams › GATE › Engineering Mathematics

Let X and Y be finite sets and $f: X \to Y$ be a function. Which one of the following statements is true?

1. For any subsets A and B of X, $|f(A \cup B)| = |f(A)| + |f(B)|$
2. For any subsets A and B of X, $f(A \cap B) = f(A) \cap f(B)$
3. For any subsets A and B of X, $|f(A \cap B)| = \min\{|f(A)|, |f(B)|\}$
4. For any subsets S and T of Y, $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$

Correct answer: For any subsets S and T of Y, $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$

Solution

For any element x in X, x belongs to $f^{-1}(S \cap T)$ exactly when f(x) belongs to both S and T. That is equivalent to x belonging to both $f^{-1}(S)$ and $f^{-1}(T)$. Hence the equality holds.

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