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Math Help - Set Problem

  1. #1
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    Set Problem

    Let A, B and C be three non-empty subsets of a set X and f be a function from X to a set Y (i.e.,f:X→Y).Given some arbitrary D⊆X,by f(D) we denote the set {y ∈ Y |y = f (x) for some x ∈ D}. Prove the following: (a) If A ∩ B ̸= ∅, then f(A ∩ B) ⊆ f(A) ∩ f(B);
    (b) f(A ∪ B) = f(A) ∪ f(B).


    Been having real problems with this problem. I have just started to study martix algebra so go easy if this might be simple.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hello123 View Post
    Let A, B and C be three non-empty subsets of a set X and f be a function from X to a set Y (i.e.,f:X→Y).Given some arbitrary D⊆X,by f(D) we denote the set {y ∈ Y |y = f (x) for some x ∈ D}. Prove the following: (a) If A ∩ B ̸= ∅, then f(A ∩ B) ⊆ f(A) ∩ f(B);
    (b) f(A ∪ B) = f(A) ∪ f(B).


    Been having real problems with this problem. I have just started to study martix algebra so go easy if this might be simple.
    What does f\left(A\right)f\left(B\right)=f\left(AB\right) even mean? Cartesian product?

    For teh second one merely note that if x\in f\left(A\cup B\right)\Leftrightarrow f^{-1}(x)\in A\cup B\Leftrightarrow f^{-1}(x)\in A\text{ or }f^{-1}(x)\in B\Leftrightarrow x\in f(A)\text{ or }x\in f(B)\Leftrightarrow f(x)\in\left(f(A)\cup f(B)\right)
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