(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose f is a function with sets A and B.

1. Show that:

[tex]I_{f} \left(A \cap B\right) = I_{f} \left(A\right) \cap I_{f} \left(B\right)[/tex]

Inverse Image of F (A intersects B) = Inverse Image of F (A) intersects Inverse Image of B.

2. Show by giving a counter example that:

[tex]f\left(A \cap B\right) \neq f\left(A\right) \cap f \left(B\right)[/tex]

F (A intersects B) does not equal F (A) intersects F(B)

2. Relevant equations

Knowledge of Sets and Inverse Images

3. The attempt at a solution

1.

Let c be an element of [tex]I_{f} \left(A \cap B\right)[/tex].

By the definition of [tex]I_{f} \left(A \cap B\right) [/tex], there is a [tex]d\in(A \cap B)[/tex] so that [tex]I_{f}(d)=c[/tex].

Since, [tex]d\in(A \cap B)[/tex], [tex] d \in A & d \in B[/tex]. Since [tex]d\inA, I_{f}(d)\in I_{f}(A)[/tex]. This follows alongside [tex]d\inB, I_{f}(d)\inI_{f}(B)[/tex].

Since [tex]I_{f}(d)=c \in I_{f}(A) [/tex] and [tex]I_{f}(d)=c \in I_{f}(B), c = I_{f}(A)\capI_{f}(B)[/tex].

Thoughts? Also would I need to show that the [tex]I_{f}(A)\capI_{f}(B) \in I_{f} \left(A \cap B\right)[/tex] to show true equality?

2.

[tex]f\left(A \cap B\right) \neq f\left(A\right) \cap f \left(B\right)[/tex]

I'm thinking either the absolute value function or a square function of some sort would show that it is not equal. Though, I'm not sure how to proceed with depicting the counter example.

Sincerely,

NA

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# Homework Help: Inverse Images and Sets (union & intersection)

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