Understanding A, B, and C in the Universal Set: Proofs and Counterexamples

[dhillon]

urgent help needed please, thanks

Homework Statement

Let A, B and C be any sets inside our universal set U. Decide whether each of the following statements is true or false. Justify your answers by giving a proof or a counterexample as appropriate.

a) A⊆B if and only if B'⊆A'

b) If A∩C ⊆ B∩C then A⊆B

Homework Equations

X \ Y sets of elements in X but not Y. Y doesn't have to be a subset of X however if it is then X \ Y is the compliment of Y in X

The Attempt at a Solution

this is how i did it however i don't think its right hence i need your help thanks

a) with these 2 venn diagrams it shows that this statement is false as the elements don't have to be in set A or B , please see the attatched file for the diagrams , I am not sure please help thank you

b) With this venn diagram i think it shows the statement is true, this is as the set A has the the same elements that are also in set B therefore the statement is true. please help me out on this , see the attached document (i named the doc 2A for the first part n 2B for the second) thank you, please help

 

[verty]

Could you make your proofs/counterexamples clearer? They are very vague. Be as precise and descriptive as you can. They are also not entirely correct.

 

[dhillon]

hey thanks for reply to the post,

im not sure on how to answer this question but i was reading a book called 'mathematical communication' and i was reading about the use of venn diagrams and so to try and answer this question i tried applying that concept. I was unsure if this was right any help on this and i would be gratefull. I have to submit my cw by monday and the thing is i know i have left it all last minute but i know excuses but this week i was resting as i just came back from the hospital. i have 5 more questions to do and I am trying to them as i read the book. please help, thank you

 

[verty]

You know what I've asked you to do and you've not done it, so I'm waiting for you to do it. Thank you. (I am trying to help.)

Let me just add, I know you can make those arguments clearer, and doing that may make things much more understandable.

 

[verty]

I'm sorry, please ignore what I've said in this thread. I don't think I can help, you need to put effort in and I'm just going to say it, those Venn diagrams were not convincing, the arguments were not convincing, and there's no one area I can give advice on, exept to say that none of it was particularly good.

Read your book again, try prove them again from scratch, then return.

 

[dhillon]

ok sorry,

basically for the part a) i think that with the diagrams drawn and the higlighted part of it mean that shows that the two diagrams are not in any way related. so if in the first part (1st diagram)elements are only in set A which is in Set B also and in the 2nd diagram no elements are in either set A or set B then how can that statement be 'if and only if' as in elements could still be in in set A without the second part of the staement needed, I am not sure this makes sense I am really sorry i know think I am wrong sorry

 

[verty]

-- Sorry, I should have read what you said just now more carefully. I see that you are misunderstanding what the second half of part A means, which is why you are having trouble with it. So read the rest of this post but focus on what the second part of part A is saying --

I think those diagrams of part A are related because both have A and B, and both are expressing a criterion about the size of A and B. The first half says that A is no larger than B, in the sense that anything in A is in B, and the second half says that the complement of B is no larger than the complement of A, in the sense that anything not in B is also not in A. So now I ask, are these two halves saying the same thing or not? Venn diagrams can help because they make the sizes of the sets clear, but a Venn diagram by itself is not a proof, and without a good argument it is not a proof. You need to provide a convincing argument so that no one can doubt that they are the same or that they aren't.

So to prove something, you must work hard to understand what is being said, else you will find it difficult to be convincing.

Work on part A for now until you understand it well and can convince someone that it is true or that it is false. And of course if there is something particular about the question you don't understand, like perhaps what "if and only if" means, then ask about that and I'm sure you'll get an answer.

-- and here is a suggestion, think about small sets of numbers A and B, like {1} and {1,2} and then think about what the complements of those sets are, and whether the second part is true for them --

 

[verty]

I think it would be helpful if I give a concrete example:

Suppose A = {Bart}, B = {Bart, Lisa}. Now A ⊆B because every member of A is in B.
A' = {Homer, Marge, Lisa}, B' = {Homer, Marge}. Now B' ⊆A' says that every member of B' is in A'. Is it true?