Discrete Mathematics - Operations with sets

[johnstobbart]

I apologize for the repost, but I had no replies to my previous post. I figured that I didn't put down a good enough attempt of a solution. I will try to explain what I did in more detail. I have read the rules for the forum, but if I'm still doing something wrong, please tell me. I want to clear a few things up involving operations with subsets of sets (I'm not sure if this is a more correct term, an example would be Set B below).

Homework Statement

U = {1, 2, 3, 4, 5, ∅, {1}},
A = {1, 3}
B = {{1}, 1}
C = {2, 4}
D = {∅, 1, 2}

Homework Equations

3 of the questions below are questions that I am asking myself. One of them is an example from a textbook.

We have the following sets:

1. D + B (with + meaning symmetric set difference)
2. B - D (with - meaning set difference)
3. A' (complement of A)
4. Is {∅, {1}} ∈ U? Why?

The Attempt at a Solution

Number 1:

D + B (with + meaning symmetric set difference)
{∅, 1, 2} + {{1}, 1}
D + B is putting the sets together without combining equal elements.

I am not too sure what to do with the {1}. I'm guessing that {1} and 1 are not the same.

When I perform the operation, I get: {∅, {1}, 2}

Number 2:

B - D (with - meaning set difference)
{{1}, 1} - {∅, 1, 2}

Set difference means that I should remove the like elements that appear in both B and D, and write what is left of B, so I should remove the 1 in set B and the answer will be what is left over.

I think the {1} would stay as there is not another {1} in D.

I get this as the solution:
{{1}}

Number 3:

A' (The complement of A)
The complement of A would be all the elements in the universal set that do not appear in A. I should remove all the elements within A that appear in U and get my answer.

So, I get: {{1}, 2, 4, 5, ∅}

However, to my knowledge, {1} is a subset of U. Can a subset also be an element of a set? I have a feeling that it can't, but I can't put it into words. If my feeling is right however, that would mean that the complement of A' is {2, 4, 5, ∅}.

Number 4:

Is {∅, {1}} ∈ U? Why?
U = {1, 2, 3, 4, 5, ∅, {1}}

I know that ∅ is normally a subset of every set, but because it's included within U, that makes ∅ an element of U. {1} is also included as a part of U, but it is a subset of U. Would that not make it an element as well?

I am guessing that the set {∅, {1}} is not an element of U. To be an element of U, U would have to be represented as: {1, 2, 3, 4, 5, ∅, {1}, {∅. {1}}}
Is that correct?

I have tried my best fleshing out my questions and trying to understand. I hope that someone can help me.

Thank you.

 

[chiro]

Hey johnstobbart and welcome to the forums.

All your answers look right with the exception of Number 3: What you wrote first is correct but then you reasoned a different answer later which is not correct. {1} is different from 1 as elements, and the way you treated it the first time was the proper way.

As you have pointed out, the elements have to be identical in their definition for set operations and this includes whether they are sets themselves like {1} or even {} (the empty set).

 

[johnstobbart]

Thanks for the welcome and thank you very much for your time, Chiro. I think I get it now.