(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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Inverse Images and Sets (union & intersection)

**Physics Forums | Science Articles, Homework Help, Discussion**