Set theory

Question

Let $f: A \to B$ be a function and $S$ and $T$ be subsets of $B$. Consider the following statements about image (range) :

$S1:\quad f^{-1}(S \cup T) = f^{-1}(S) \cup f^{-1}(T)$
$S2:\quad f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$

Which of the following is correct?

A) only S1 is true

B) only S2 is true

C) Both S1 and S2 is true

D) Neither S1 nor S2 is true

Comments

Both are true? I doubt about $1^{st} $one, though thinks it must be true?

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Both seem to be true.

I have taken bijection about integers to prove this.

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Ashwin Kulkarni  Sir , will it always  be  true that S U T will always give B ?

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 Pawan Kumar 2 No that's not always true

Answer 1

A = {1,2,3,4}
B = {1,4,9,16}
S = {1,4}
T = {4,9}

for S1:  f⁻¹( S∪T ) = f⁻¹(S) ∪ f⁻¹(T) 

$\Rightarrow$  f⁻¹(1,4,9) = f⁻¹(1,4) ∪ f⁻¹(4,9)
$\Rightarrow$ {1,2,3} = {1,2} ∪ {2,3}

$\Rightarrow$ {1,2,3} = {1,2,3}

hence S1 is true

for S2: f⁻¹(S∩T) = f⁻¹(S) ∩ f⁻¹(T)

$\Rightarrow$   f⁻¹(4) = f⁻¹(1,4) ∩ f⁻¹(4,9)

$\Rightarrow$  {2} = {1,2} ∩ {2,3}
$\Rightarrow$ {2} = {2}
hence S2 is true

Comments

Nice explanation....