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# HELP - Venn Diagrams watch

1. Not overly sure on this so thought I'd ask here to see if someone could give me some guidance.
------------------------

For any two non-empty sets A and B drawn from the same universe, U,
let Z be the set defined as
Z = (A AND ~B) OR (~A AND B)

Note that Z is the symmetric difference of A and B.

Through the use of Venn diagrams show the following:

(i) Z could equal ( 'O' with a '/' through it, also shown as {})

(ii) Z could never equal B

(iii) Z can equal U

(12 marks)

---------------

The first thing I've done for this question is worked out what Z looks like in Venn Diagram form

I then guessed that the answer to part (i) is

But I'm not sure of this because it's looking for where 'Z could equal an empty set' and I'm not sure if this is the area which is outside of the 2 venn diagrams.

The second part I'm guessing is this shaded area.

However when I look back what Z is I'm a bit confused as to whether or not I'm doing this question correctly.

Perhaps I just don't understand the question but it would be helpful if someone could give me some advice with this question.
2. (Original post by mos182)
Perhaps I just don't understand the question but it would be helpful if someone could give me some advice with this question.
You're not being asked to draw the standard Venn diagram for two sets each time.

You're being asked to choose an appropriate A and B, such that Z matches the criteria above. You need to choose how A and B related to each other.

As examples, one set could be the universal set, or they might not "overlap", etc.
3. (Original post by ghostwalker)
You're not being asked to draw the standard Venn diagram for two sets each time.

You're being asked to choose an appropriate A and B, such that Z matches the criteria above. You need to choose how A and B related to each other.

As examples, one set could be the universal set, or they might not "overlap", etc.
Yes I know but they ask you to show the answers using a Venn diagram and since there's only A and B it's easiest just to show it using the standard Venn Diagram.

The question doesn't make sense to me. I've never experienced a question like this before. They give me a equation as to what Z is then expect me to illustrate what Z is on a Venn Diagram with these questions.
4. (Original post by mos182)
Yes I know but they ask you to show the answers using a Venn diagram and since there's only A and B it's easiest just to show it using the standard Venn Diagram.

The question doesn't make sense to me. I've never experienced a question like this before. They give me a equation as to what Z is then expect me to illustrate what Z is on a Venn Diagram with these questions.
Yes, I see your point. If we consider a Venn diagram to contain all possible combinations of A,B then trying to demonstrate the various criteria, doesn't make sense.

Only if we relax the restriction of showing all possible combinations, and use Venn-like diagrams is it feasible, IMHO.

Perhaps someone else may have a useful comment, or check back with your lecturer, are the only things I can suggest.
5. (Original post by ghostwalker)
Yes, I see your point. If we consider a Venn diagram to contain all possible combinations of A,B then trying to demonstrate the various criteria, doesn't make sense.

Only if we relax the restriction of showing all possible combinations, and use Venn-like diagrams is it feasible, IMHO.

Perhaps someone else may have a useful comment, or check back with your lecturer, are the only things I can suggest.
I can't ask the lecturer because I'm not back until Monday at Uni. Currently off for Easter and he's left us with an assignment. We've got a Facebook group and I don't think anyone on there knows how to do it.

I originally thought the question was asking me basically this.

(ii) Z could never equal B

So basically this means

(A AND ~B) OR (~A AND B) =/= B

Which makes me think that I just show only A shaded. Like this.

Or perhaps show it as A shaded in and A and B not over lapping if that makes sense?

Something else which confuses me is the statement below.

'For any two non-empty sets A and B drawn from the same universe, U'

I'm guessing this means basically everything which isn't A or B is the universe U?

Another thing that doesn't make sense is how could Z equal an empty set?
6. (Original post by mos182)
...
In each case you want to show Z.

So, yes for ii), I'd do two non-overlapping circles. In that situation, Z cannot equal B, as it contains A as well.

'For any two non-empty sets A and B drawn from the same universe, U'

I'm guessing this means basically everything which isn't A or B is the universe U?
The universe will include A, and B. This is your outside oblong in the Venn diagram.
7. (Original post by ghostwalker)
In each case you want to show Z.

So, yes for ii), I'd do two non-overlapping circles. In that situation, Z cannot equal B, as it contains A as well.

The universe will include A, and B. This is your outside oblong in the Venn diagram.
That's what I thought.

So for each case, you just label whatever part you think is Z.

I'm not sure what is the answer for (i), it's really confused me. How can you show that Z is an empty set? Unless you just show that there is no A or B and it's just the universe U?
8. (Original post by mos182)
That's what I thought.

So for each case, you just label whatever part you think is Z.
Agreed.
See below. Post #12

I'm not sure what is the answer for (i), it's really confused me. How can you show that Z is an empty set? Unless you just show that there is no A or B and it's just the universe U?
Well you're told that A,B are non-empty, so you'll need a different solution.

Hint: If you break down Z into its two main components, what has to be true for each of them to be the empty set?
9. (Original post by ghostwalker)
Agreed.

Well you're told that A,B are non-empty, so you'll need a different solution.

Hint: If you break down Z into its two main components, what has to be true for each of them to be the empty set?
So both A and B are true instead of only 1 of them being true?
10. (Original post by mos182)
So both A and B are true instead of only 1 of them being true?
A and B are sets, they're not "true". It's the relationship between them that has to exist for A\B to be the empty set. And similarly the other way around.
11. (Original post by mos182)
So for each case, you just label whatever part you think is Z.
Although I previously agreed with this, I didn't pay close enough attention.

You don't need to label Z, in fact you can't neessarily. You just need to set up A and B, such that Z will satisfy the given criteria.
12. (Original post by ghostwalker)
A and B are sets, they're not "true". It's the relationship between them that has to exist for A\B to be the empty set. And similarly the other way around.
What if one of them was missing? Would it be an empty set? Or if it was B and C?

Really puzzled by this
13. (Original post by mos182)
What if one of them was missing? Would it be an empty set? Or if it was B and C?

Really puzzled by this
Just consider A and B, neither of which is the empty set, but A\B is the empty set. What does this imply?
14. (Original post by ghostwalker)
Just consider A and B, neither of which is the empty set, but A\B is the empty set. What does this imply?
what does '\' mean?
15. (Original post by mos182)
what does '\' mean?
If means the set of elements that are in the first set (A) and not in the second set (B).

It's the standard notation for the set-theoretic difference.

I assumed you were aware of it, as that's what you're working with. But clearly you've not yet covered it.
16. Not yet, not sure if we cover that. We've got a lack of examples which is frustrating.

It doesn't tell you the set of elements that are in either sets?
17. (Original post by mos182)
Not yet, not sure if we cover that. We've got a lack of examples which is frustrating.

It doesn't tell you the set of elements that are in either sets?
You don't need to know.

A\B consists of the elements that are in A and not in B. But we want this to be the empty set.

So, there are no elements satisfying being "in A and not in B".

This tells you something about the relationship between the sets A and B.

Not sure how else to phrase it without just telling you, which I'll do as a spoiler in the next post. But if would be better if you can discover it yourself.
18. (Original post by ghostwalker)
So, there are no elements satisfying being "in A and not in B".
Ah A is a subset of B?
19. (Original post by mos182)
Ah A is a subset of B?
Bingo!!!!

So from the first part of the definition of Z you've worked out A is a subset of B.

Now what about the second part?
20. (Original post by ghostwalker)
Bingo!!!!

So from the first part of the definition of Z you've worked out A is a subset of B.

Now what about the second part?
Finally I'm getting something right

B is a subset of A?

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