Question

Let X and Y be finite sets and f : X $\to$ Y be a function. Which on eof the folowing statements is TRUE?

• A. For any subset A and B of X.
  |f(A $\cup$ B)| = |f(A)| + |f(B)|
• B. f(A $\cap$ B) = f(A) $\cap$ f(B)
• C. |f(A $\cap$ B)| = mim{|f(A), f(B)|}
• D. $f^{-1}$ (S $\cap$ T) = $f^{-1}$(S) $\cap$ $f^{-1}$(T)

Answer 1

The correct option is D $f^{-1}$ (S $\cap$ T) = $f^{-1}$(S) $\cap$ $f^{-1}$(T)

Example


Let A = {1, 2}, B = {3, 4}
(a) |f(A $\cup$ B)| = |f(A)| + |f(B)|
$\Rightarrow$ 3 = 2 +2 is false
(b) f(A $\cap$ B) = f(A) $\cap$ f(B)
$\Rightarrow$ $\varphi$ = {a} is false
(c) |f(A $\cap$ B)| = mim{|f(A), f(B)|}
$\Rightarrow$ 0 = min (2, 2) = 2 is false
(d) Let S = {a, b} T = {a, c}
$f^{-1}$ (S $\cap$ T) = $f^{-1}$(S) $\cap$ $f^{-1}$(T)
$\Rightarrow$ {1, 3} = {1,2, 3} $\cap$ {1, 3, 4}
$\Rightarrow$ {1,3} = {1, 3} is true