1. ## Set Theory Help

Hi,

I'm currently going through past exams papers as part of my revision for my upcoming exams, however the past papers that I am using do not include answers, so I'm not sure if I'm on the right track or not.

The question that I am working on is:

Write down the letter corresponding to one true statement among the following:

(A) {A,B}\{A,B,C}={A,B}
(B) {A,B}\{A,B,C}={C}
(C) {A,B}\{A,B,C}={A,B,C}
(D) {A,B}\{A,B,C}={} (i.e. the empty set)
(E) None of the above statements is true

It has been a while since I did set theory as it was the beginning of the year and it was an area I struggled with, but am I right in saying that the correct answer is D?

Any advice on this will be appreciated and if anybody can recommend any websites that contain example questions with answers, then this will be appreciated too.

2. Hello,

Yes D is the correct answer !
C\D = {elements of C that don't belong to D}
Since C is included in D, there's no such element, hence the answer is the empty set !

3. Originally Posted by Moo
Hello,

Yes D is the correct answer !
C\D = {elements of C that don't belong to D}
Since C is included in D, there's no such element, hence the answer is the empty set !
Thanks for the response. I don't quite understand the second bit of your response, where you have mentioned that C is included in D and therefore there is no such element. Once I've got my head around this properly I think I'll be fine.

4. C\D is the set of all elements of C not belonging to D.
Buf if C is included in D, all the elements of C are in D (by definition of inclusion). Hence C\D=empty set

5. Originally Posted by Gall1987
Thanks for the response. I don't quite understand the second bit of your response, where you have mentioned that C is included in D and therefore there is no such element. Once I've got my head around this properly I think I'll be fine.
Another way of looking at it. Think of the subtraction operation. (This is not a perfect analogy, but it works.)
{A, B}\{A, B, C} = {A, B} - {A, B, C} <-- I've seen texts using this terminology, actually.

A - A means A is removed.

B - B means B is removed.

-C means, well, don't do anything with C.

What's left of {A, B}? Nothing. Both A and B have been removed. So {A, B}\{A, B, C} is the empty set.

-Dan

6. The defintion of $\displaystyle X\setminus Y=X-Y\text{ is }X\cap Y^c$.
That is, $\displaystyle X$ intersect $\displaystyle Y$ complement.
So nothing in $\displaystyle Y$ can be in $\displaystyle X\setminus Y$.