You are Here: Home >< Maths

1. If is a one to one function of the set to the set and then

If then there exists so that . Then or

So then so and as well so there exists with and is a one to one so surely with contradicting

So the contradiction proves that and in turn is true.

Is this correct?
2. (Original post by AishaGirl)
If is a one to one function of the set to the set and then
What is supposed to mean here? I've never seen that notation used in this context.

I think you have a typo here: showing is hardly an achievement!
3. (Original post by DFranklin)
What is supposed to mean here? I've never seen that notation used in this context.

I think you have a typo here: showing is hardly an achievement!
Oh yes my bad, I meant to say it proves that

is the symmetric difference, some books use the symbol. Sorry I should have clarified that.
4. (Original post by AishaGirl)
Oh yes my bad, I meant to say it proves that

is the symmetric difference, some books use the symbol. Sorry I should have clarified that.
OK, I perhaps should have guessed, but is used for a lot of other things at university and so it's not usual for it to be used for symmetric difference at that level.

(Original post by AishaGirl)
If then there exists so that . Then or

So then so and as well
I don't see why this line follows. (I also don't see where you are "starting" your contradiction; it's usual to say something like

"assume (for contradiction) that X is true. {...your argument...}, but this gives a contradiction and therefore it cannot be the case that X is true".

It's important to have the preamble, because (since you want a contradiction) the initial assumption is wrong, and if you're going to write something you know is wrong, you need to make it clear what the reason is). You don't have to actively say "for contradiction" (though I would do so at A-level), but you do need to make it clear it's an assumption, not something you think is actually true.

I'm not sure I'd prove this by contradiction anyhow, rather show that if then and vice-versa.
5. DFranklin

If I do it in the other direction and suppose then or then such that Suppose that also then again contradicting the fact that

So proves that it the subset is true? I don't see the problem with using contradiction, are you just saying it's a lengthier way of doing it or rather that it's bad practice?

What is a simpler way of doing it?

Thanks.
6. (Original post by AishaGirl)
DFranklin

If I do it in the other direction and suppose then or then
Why do you say that ? It clearly doesn't have to be.

I don't see the problem with using contradiction, are you just saying it's a lengthier way of doing it or rather that it's bad practice?
Actually I'll kind of take that back; I think you'll probably have some contradiction argument somewhere. It's just that when you say "I'm going to prove X by contradiction", you usually mean "I'm going to assume X isn't true and derive a contradiction", whereas here it's more you're going to get some way down the road and then say to finish "I'd like Y to be true, and it must be true, because otherwise we get a contradiction".
7. (Original post by AishaGirl)
What is a simpler way of doing it?
I've only skimmed the previous posts so I'm not sure if your contradiction approach will work, but DFranklin has already outline another approach.

In fact, it is the standard approach. If you want to prove that where A and B are sets, then you have to show that since that is the definition of equal sets.
8. (Original post by atsruser)
I've only skimmed the previous posts so I'm not sure if your contradiction approach will work, but DFranklin has already outline another approach.

In fact, it is the standard approach. If you want to prove that where A and B are sets, then you have to show that since that is the definition of equal sets.
So, actually I think this is what she's trying to do; the contradiction comes later. "As you know...", if you're trying to prove things with the symmetric difference it's pretty common to get to a point where you want to say x is in M or N, but not both, and so end up with something like, "suppose X is in M and M, then ... which is a contradiction".

I have to say that I kind of hate "fundamental" questions about set operations, because it always feels to me like "it's obvious", and any step-by-step proof doesn't actually make the truth of the statement clearer, rather the opposite. This probably indicates some deep flaw in my character...

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: January 31, 2017
Today on TSR

### The TSR A-level options discussion thread

Choosing A-levels is hard... we're here to help

### University open days

• Heriot-Watt University
School of Textiles and Design Undergraduate
Fri, 16 Nov '18
• University of Roehampton
Sat, 17 Nov '18
• Edge Hill University
Faculty of Health and Social Care Undergraduate
Sat, 17 Nov '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams