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GATE 2014 CS (CS3) – Question 12

Engineering Mathematics · Discrete Mathematics: Sets, Relations, Functions, Partial Orders and Lattices · 1 mark · Multiple choice

Let X,Y be finite sets and $f:X\to Y$. Which statement is TRUE?

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

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Correct answer: (D) For all $S,T\subseteq Y$, $f^{-1}(S\cap T)=f^{-1}(S)\cap f^{-1}(T)$.

Explanation

An element belongs to f⁻¹(S∩T) exactly when its image belongs to both S and T. Hence f⁻¹(S∩T)=f⁻¹(S)∩f⁻¹(T), **D**. Direct images need not preserve intersections because different inputs may share an image.