I'm currently taking this college calculus course, and this exercise has stumped me. It is in German, but hopefully what it's asking is fairly clear.
To summarize, the problem first asks me to prove that the preimage of an intersection of a family of sets is equal to the intersection of a preimage of a family of sets, where f: A -> B and Ui is a family of subsets of B. To do this, I just gave the general example of x being an arbitrary element of A, which means that x is an element of the preimage of the intersection, which means for all values of i f(x) is an element of Ui, which means x is an element of the intersection of the preimage of Ui.
Image of my answer cause I can't figure out MathJax:
Then the problem asks to show that for a family Vi, a subset of A, an image of the intersection of a family of sets is not in general equal to the intersection of an image of a family of sets.
Yet, for every example formula and sets I can conjure up, I always seem to prove that the two sides are in fact equal. No amount of coaxing and complaining on Bing or ChatGPT causes them to bring an example up either – they just keep giving examples where the two sides are equal.
So, is this a typo from my professor and this expression actually is generally correct, or am I missing something? What family of sets and functions would make these not equal?
Thank you!
Best Answer
Guide:
Let $V_1$ be the set of positive integers, let $V_2$ be the set of negative integers. By designing my set to be so, note that $V_1 \cap V_2 =\emptyset$ and hence $f(V_1 \cap V_2)=\emptyset$.
Now, can you design your function such that $f(V_1)$ and $f(V_2)$ has non-empty intersects?