Thread: Sets

Let A,B,C be sets.

(i)Which of the following are always true?

1). A \ ( B \ C) = ( A \ B) U C
2). A \ (B U C) = (A \ B) \ C
3). A \ (B <intersection> C)= (A \ B) U (A \ C)

I have the first one as not always true and the second and third as always true. I just wanted someone to check this because I think i'm doing them wrong.

Is there a difference between A \ B \ C and (A \ B) \ C?

Sorry I don't know how to get unions and intersections to work so I used a U for the union and wrote <intersection> for intersections.

Hello,

Originally Posted by Showcase_22

I have the first one as not always true and the second and third as always true. I just wanted someone to check this because I think i'm doing them wrong.

Is there a difference between A \ B \ C and (A \ B) \ C?

Sorry I don't know how to get unions and intersections to work so I used a U for the union and wrote <intersection> for intersections.

You can rewrite :


It's \cup for union and \cap for intersection

A \ B \ C

I don't think this is a correct writing, unless you can prove (A\B)\C=A\(B\C)

I don't think this is a correct writing, unless you can prove (A\B)\C=A\(B\C)

But surely if the set is A not B not C then there would just be A regardless of where you put brackets?

Where did this come from? I'm trying to think of it as a Venn diagram but I can't see what it would look like.

ohhh, so A \ B \ C is not the same as (A \ B) \ C. That seems weird! does "\" mean "not"?

My foundations lecturer introduced "¬" that apparently also means not. It seems strange that he would introduce two different symbols that apparently mean the same thing! (he also did the same for ^ and and it's counterpart with the union. Is there a difference between all this notation???)

Originally Posted by Showcase_22

But surely if the set is A not B not C then there would just be A regardless of where you put brackets?

Why ?

Where did this come from? I'm trying to think of it as a Venn diagram but I can't see what it would look like.

Hmm well, it's the definition oO
How do you define ?
It's the set of elements that are in A but not in B, that is to say they are in A and in

Hang on, I think i'm missing something.

"\" means "not" doesn't it?

For example, "A \ B" means "A not B".

I'm pretty sure that's what our lecturer told us. =S

Originally Posted by Showcase_22

Hang on, I think i'm missing something.

"\" means "not" doesn't it?

For example, "A \ B" means "A not B".

I'm pretty sure that's what our lecturer told us. =S

Hmmm "not A" usually stands for , it affects only one set.
Here, \ is a binary operator, that is to say it deals with two sets.

If you prefer, it's "A and not B" and we can refer to

elements that are in A but not in B

Okay then. Going back to the first question:

A \ (B \ C)=A \ (A \ B)

(I used d instead of c since C was already a set).

The second question would become:

Need to show: A \ =(A \ B) \ C
(A \ B) \ C= \ C

Would I need another identity to show that it's true? (if it's true, I think it is). Truthfully, I would much rather remember identities and try to get one side to equal the other side. Venn diagrams are rather limited.

#3 is correct.


This set operation is known as setminus.
The are two notations in use: