Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

How to prove this set identity?

I have to use an element-wise approach. So to prove two sets are equal you have to show they are subsets of each other. So:

How do I show that A is a subset of the left-hand side?

Last edited by LuisRodg (2008-04-22 04:38:30)

Dragonshade

Member

Registered: 2008-01-16

Posts: 147

Re: How to prove this set identity?

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: How to prove this set identity?

I've not done any of this.. but surely it's obvious from a Venn Diagram?  (not a particularly helpful comment )

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: How to prove this set identity?

intersection of A and (the union of A and B)

ehm, havnt really done any set theory, let me take a try anyway using defitionos of union and intersection

lol, that looks so terrible im ashamed, nevermind; it makes sense to me. reasoning for last step, in both the bracketed parts, there is an x ∈ A, so if x isnt in A, then both sides are false, if x IS in A, then the left side is true, and whether the right side is true or not is irrelevant since its an OR inbetween them

Last edited by luca-deltodesco (2008-04-22 04:51:06)

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LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to prove this set identity?

@Luca

I looked on the back of the book and the answer they show is the same as your first attempt:

So since it shows both parts are subsets of each, they are equal.

@Daniel

Venn diagrams are not a tool to proof in Set Theory. Venn diagrams constitutes a way to visualize but you cannot prove anything with them.

Last edited by LuisRodg (2008-04-22 05:05:01)

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to prove this set identity?

Prove:

So how do I prove the second one? They both make sense but im just unsure how to go on about the proof. Do I need to work from the right side?

Last edited by LuisRodg (2008-04-22 05:22:45)

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: How to prove this set identity?

same argument really

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LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: How to prove this set identity?

Ok got it.

For exam-taking purposes, how do you decide in which side to take the element approach? Does it depend if we are talking unions or intersections?

Last edited by LuisRodg (2008-04-22 05:29:04)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: How to prove this set identity?

In general, if you want to prove A subset of B, let x be in A and show that x is in B.  If you want to show equality, you must show A is a subset of B and B is a subset of A.

The easiest way to do these proofs is to expand upon what it means for x to be in a set.  For example, if x is in A intersect B, then x is in A and x is in B.  On the other hand, if x is in A union B, then x is in A or x is in B.

Stating what you know and what you need to show make the problems very easy.  On the last one you posted:

You know: x is in A union B, so x is in A or x is in B.
You need to show: x is in A union B union C, meaning show x is in A or x is in B or x is in C.

If x is in A or B, then certainly x is in A or B or C.  And you're done.

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JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How to prove this set identity?

Prove:

For the first one, let

.

Then

For the second one, let

.

Then

Scorpy

Member

Registered: 2011-10-18

Posts: 1

Re: How to prove this set identity?

Hello everybody

I'm trying really hard to understand these proof methods, but still haven't clear it at all. So could someone explain me the following:

A is a subset of B if and only if A intersection B complement is empty set?

I think that it's not true because of the A intersection B complement ->x is in A, but not in B, so that's empty set. And we have the statement A is subset of B, so x is A and in B.

Thanks

Last edited by Scorpy (2011-10-18 05:56:22)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,803

Re: How to prove this set identity?

hi Scorpy,

Welcome to the forum! 

I like to make pictures to explain maths.

Have a look at the Venn diagram below.

The blue shaded region is {A but not B}

If it is empty imagine it is rubbed out.

Then all of A fits inside B

And if you start by saying "All of A fits inside B" then the bits of A that are not in B don't exist.

It's not a rigorous proof but it might help you to see what you are trying to show.

Bob

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