Prooving fundamental properties of functions of families of sets ( Title suggestions welcome)

Asked Jul 29 '16 at 18:08

Active Jul 29 '16 at 18:33

Viewed 27 times

I am an online learner and I was faced with the following conceptual understanding gaps in proving the following: $$f(\bigcup_{i\epsilon I}A_i)=\bigcup_{i\epsilon I}f(A_i)$$ and ( drumroll please!) $$f(\bigcap_{i\epsilon I}A_i)\subseteq\bigcap_{i\epsilon I}f(A_i)$$

If some illustrative examples are provided, that would be really nice. Also, answer with caution, I am a sensitive person who is using this website for the second time only.

Thanks in advance.

edited Jul 29 '16 at 18:33

Asaf Karagila

393,674

asked Jul 29 '16 at 18:08

Karan Kumar

The best way to understand these "conceptually" is probably to try to prove them yourself. (Perhaps start with the simple case where only two sets are involved.) In particular, what goes wrong when you try to show that $\bigcap f(A_i) \subset f(\bigcap A_i)$? – Alex Provost Jul 29 '16 at 18:11
@Martin Sleziak has assembled a very useful collection of identities (with stackexchange links provided) for this type of question. See Overview of basic results about images and preimages. – Dave L. Renfro Jul 29 '16 at 19:27

Prooving fundamental properties of functions of families of sets ( Title suggestions welcome)

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