Does this proof work?

demon8991

demon8991

demon8991

Soldato
Joined
7 Jan 2003
Posts
4,458
Location
Gold Coast, Australia
(A union B) intersection (A union C) = A union (B intersection C)

I have go an eqaution and i want to proove that it works, is this correct?

Choose an aribitary X belongs to (A union B) intersection (A union C), then X belongs to A union B and X belongs to A union C.

If X doesnt belong to A then X must belong to B and C which means that X belongs to B intersection C.

Which means X belongs to A union (B intersection C) this remains true when X belongs to A.

Therefore X belongs to (A union B) intersection (A union C) implies that X belongs to A union (B intersection C) and becuase X is arbitary this is trye for every such X and so.

(A union B) intersection (A union C) is a subset of A union (B intersection C)

Doing very much the same thing you can then prove that:

A union (B intersection C) is a subset of (A union B) intersection (A union C)

And becuase they are both subsets of each other then.

(A union B) intersection (A union C) = A union (B intersection C)

is correct?

Please tell me this is right...
 
Yep, that's exactly right, as long as the part of your proof where you've said "and you can do something similar..." works out too. I'm sure it does.

Just remember the simple maxim:

Two sets A, B are equal if, and only if, "x in A => x in B" and "x in B => x in A"...

... or equivalently, A, B are equal if AcB and BcA, where 'c' means 'is a subset of'.

I also find that it helps to write AnB for 'A intersect B' and AuB for 'A union B', rather than writing the entire thing out in words. ;)
 
Ahhh brilliant im glad that it works.

Yer i was going to do that just thought people may get confused for it being another variable.
 
Out of interest does this work the exact same way for.

(A intersection B) union (A intersection C) = A intersection (B union C)
 
Arcade Fire said:
Yep, that's exactly right, as long as the part of your proof where you've said "and you can do something similar..." works out too. I'm sure it does.

Just remember the simple maxim:

Two sets A, B are equal if, and only if, "x in A => x in B" and "x in B => x in A"...

... or equivalently, A, B are equal if AcB and BcA, where 'c' means 'is a subset of'.

I also find that it helps to write AnB for 'A intersect B' and AuB for 'A union B', rather than writing the entire thing out in words. ;)
Click to expand...

Wow man, all I see is a dyslexic alphabet
 
demon8991 said:
Out of interest does this work the exact same way for.

(A intersection B) union (A intersection C) = A intersection (B union C)
Click to expand...
Yep, exactly the same way.

If 'x in AnB' then 'x in A' and 'x in B', so 'x in BuC'. Thus 'x in An(BuC)'.

If 'x not in AnB' then 'x in AnC', so 'x in A' and 'x in C'. Thus 'x in BuC', and finally 'x in An(BuC)'. So (AnB)u(AnC) is a subset of An(BuC).

You then do the same thing for the converse [i.e. to show An(BuC) is a subset of (AnB)u(AnC)] and you're done.
 
BTW demon, there are symbols for intersection and union and other set theory operators that you can post on the interwebs:

Intersection: ∩
Union: ∪

It might save you some typing in the future. ;) :)
 
demon8991 said:
Wicked so the proof is identical just with the opposite notation?
Click to expand...
Yes.

Where are you studying maths, by the way? It just occured to me that there's a very similar question to this on the first problem sheet at my uni... ;)
 
Right, a set is a collection of objects - they could be anything, like I could have the set {1,2,3} which consists of the numbers 1, 2 and 3, or I could have the set consisting of all the monkeys in the world, or the set consisting of my right hand and my left hand... it's not really that important what the objects in sets are, so we mainly talk about sets of numbers for convenience.

If you want to talk about sets in a meaningful way, to prove things about them, i.e. to do maths with them, you need a bit of terminology and notation so that everyone understands what you're talking about. [Aside: as to why you'd want to talk about sets... it's important in computer science, among other things]

If we have two sets, for example {1,2,3} and {3,4,5,6} then the UNION of those two sets is the set which contains all the members of the first one and all the members of the second one. The union of our two sets is {1,2,3,4,5,6}. Notice that you don't count the '3' twice - sets only ever have one of each object in them. There's no such thing as the set of four threes.

The INTERSECTION of two sets is the set which contains all of the members of the first one which are also members of the second one. So the intersection of our two sets is {3}, since '3' is the only thing which is in both of the sets.

It's cumbersome to talk about this in words, so we use a shorthand - if you call your two sets A and B, then the union of A and B is written as AuB, and the intersection of A and B is written as AnB.

Another thing that we need if we want to talk about sets meaningfully is a sense of when two sets are the same. The obvious way to have two sets the same is when they contain the same objects. If we want a more technical definition, then we say that two sets A and B are EQUAL when every member of A is also a member of B, and every member of B is also a member of A.

So, the question that demon has been asked is "Prove that (AuB)n(AuC)=Au(BnC)", i.e. prove that if you have three sets A,B,C then taking the intersection of B and C and then taking the union of that with A, is the same as taking the unions of A,B and A,C, and then taking the intersection of those two sets.

That was a really rushed introduction to the basic ideas behind set theory - I've often thought about writing up something more concrete (and understandable) but I doubt I'll ever get around to it.
 
Just out of interest.....

What is the reasons for spending so much time on set theory in maths ? Is it any help in the real world ? Will it help the hundreds of schoolkids who can't do simple long division / multiplication etc & have real problems with basic maths?

There were so many sub branches of maths - that I found a complete waste of time ( even though I did well at them) - and never used them again in 20 years of working.
 
divosuk said:
What is the reasons for spending so much time on set theory in maths ? Is it any help in the real world ? Will it help the hundreds of schoolkids who can't do simple long division / multiplication etc & have real problems with basic maths?
Click to expand...
As far as I'm aware, it's not taught at A Level and it's certainly not taught at GCSE... unless things have changed since my day! So I don't think it really impacts on the educations of schoolkids at all.

It's one of the most fundamental branches of mathematics, and as such it will have very little relevance to 'the real world' if by the real world you mean the everyday lives of most people. Of course, most people's lives are intricately interwoven with technology, much of which would have been impossible without the developments which came from the study of set theory in the early 1900s and beyond... not least of which is the computer revolution.

There are good reasons to study it despite that, though. Firstly, if you want to do any form of higher mathematics or computer science, then it's vital to understand at least some set theory. It's also an excellent tool for grasping the roles of logic and proof in mathematics, which will help you to understand the rest of the subject, and generally help to train your mind. So many arguments that I see on here would never have arisen if people thought just a little more logically.
 
Yer im doing it as part of my university course, finding parts of it quite confusing as its unlike any maths ive done before its much more abstract.
 
divosuk said:
Just out of interest.....

What is the reasons for spending so much time on set theory in maths ? Is it any help in the real world ? Will it help the hundreds of schoolkids who can't do simple long division / multiplication etc & have real problems with basic maths?

There were so many sub branches of maths - that I found a complete waste of time ( even though I did well at them) - and never used them again in 20 years of working.
Click to expand...

Well its a university course, set theory is used in computer science in real life.. so there is one real world application of it. There are many branches of maths that seem pointless at first like number theory, but are applicable to areas like cryptography etc.. It won't help the schoolkids who can't do long division/multiplication etc... but they won't have to deal with it ever.
 
You must log in or register to reply here.
Back
Top Bottom