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So, I was assigned a problem in my Intro Analysis course that involves proving, by induction, that the set [itex]A[/itex] minus some arbitrary number of intersections of the sets [itex]B_{j}[/itex] is equal to some arbitrary number of unions of [itex]A[/itex] minus the sets [itex]B_{j}[/itex].

I've written out a proof, but I'm not too comfortable with it. It's the notation, specifically, the notation for arbitrary intersections/unions, that I'm not 100% on - I've never taken a "Fundamentals" course, and although my professor offered a quick review of sets during the first week, he never covered this sort of stuff.

Anyway, I was hoping someone could take a look at it and help me correct any errors in the proof. Oh, and I'm new here and I've never used Latex before, so sorry in advance if this post comes out sloppy or unreadable.

1. The problem statement, all variables and given/known data

Use mathematical induction and your result from problem 1-1a to prove the statement below. (Do NOT prove it by showing that each side of the equation is a subset of the other side. Use induction.)

If [itex]A, B_{1}, B_{2}, ..., B_{n}[/itex] are sets, then [itex]A\setminus\bigcap_{j=1}^{n}B_{j}=\bigcup_{j=1}^{n}A\setminus B_{j}[/itex] .

2. Relevant equations

Result from problem 1-1a:

[itex]A\setminus(B\cap C)=(A\setminus B)\cup (A\setminus C)[/itex]

3. The attempt at a solution

We use mathematical induction.

Let P(n) be the statement [itex]A\setminus\bigcap_{j=1}^{n}B_{j}=\bigcup_{j=1}^{n}A\setminus B_{j}[/itex] .

Setting n=1, we get [itex]A\setminus B=A\setminus B[/itex], which is trivially true.

Next, we let k be an arbitrary natural number and assume P(k); we will show P(k+1).

So, we have

[itex]A\setminus\bigcap_{j=1}^{k+1}B_{j}[/itex]

[itex]=A\setminus(B_{k+1}\cap\bigcap_{j=1}^{k}B_{j})[/itex]

[itex]=(A\setminus B_{k+1})\cup(A\setminus\bigcap_{j=1}^{k}B_{j})[/itex] (from proof of 1-1a)

[itex]=\bigcup_{j=1}^{k+1}A\setminus B_{j}[/itex] .

Therefore, if [itex]A, B_{1}, B_{2}, ..., B_{n}[/itex] are sets, then [itex]A\setminus\bigcap_{j=1}^{n}B_{j}=\bigcup_{j=1}^{n}A\setminus B_{j}[/itex] .

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# Homework Help: Induction proof involving arbitrary intersections/unions

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