Solve Sets & De Morgan Homework - Introduction to Analysis 5th Ed. Gaughan

[Krovski]

Homework Statement

I'm using Introduction to Analysis 5th edition by Edward D. Gaughan.

The question is:
Prove (De Morgan)
S\([itex]\bigcap[/itex] A[itex]_{\lambda}[/itex]) = [itex]\cup[/itex](S\A)
[itex]\lambda\epsilon[/itex] [itex]\Lambda[/itex]
Where [itex]\Lambda[/itex] A and S are sets
(doesn't specify real or complex but assuming real)

Homework Equations

to prove two things equal it is enough to show they are contained within each other
union is all elements of both sets, none repeat (ie if two appears twice count it only once)
intersection is all elements common to both sets

The Attempt at a Solution

(for typing purposes I'll just write everything out in words rather than symbols, having trouble from my tablet)
I can understand that if there is some x that belongs to this set, it is clear that it belongs to S and not the intersection of A[itex]\lambda[/itex]
On the right hand side of the equation I can see that if x is not in the intersection of A[itex]\lambda[/itex] and it is in S, then the union of the two sets would mean that x is in the right hand side of the equation since it's in S.

I don't think it is enough for what my Prof. is looking for. Any thoughts?

 

[jmjlt88]

Krovski, perhaps we can start off a little smaller.

Let A,B, and C be sets.

For three sets, the DeMorgan's Law in question states:

A-(B ∩ C) = (A - B) ∪ (A - C).

*I am using A-B to denote {x| x is in A and x is not in B}

Prove that first. Then, perhaps solving this will give us a better intuition for arbitrary unions and intersections.