Subset Set Theory Help

thunc14

Junior Member
Joined
Nov 15, 2017
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I'm having a couple confusions on this problem set. I got the hang of a lot of the notation and concepts, but I would really appreciate a look over to make sure I'm on the right track. My answers are in red, and problems that I'm a bit iffy on are in blue. Thanks in advance :eek:.

1. Given B = {m, n, p}, consider if the following statements are true or false.
a. m ∈ B
True
b. B ⊂ {m, n, p}

True
c. m ⊂ B

False
d. {m} ∈ B

False
e. {m} ⊂ B

True
d. ∅ ⊂ B

True
2. Given A = {2, {4, 5}, 4}, consider if the following statements are true or false.

a. {4, 5} ⊂ A

False
b. {{4, 5}} ⊂ A

True
c. {5} ⊂ A

False
d. {4, 5} ∈ A

True
e. {5} ∈ A

False
f. 5 ∈ A

False
3. Find all the subsets of the following sets

a. {1}

{1}, ∅

b. {a, b}

{a,b}, {a}, {b}, ∅

c. {{1, {3, 5}}}

{{1, {3, 5}}, {{1}}, {{3, 5}}, ∅

d. {1, 3, 5}

{1, 3, 5}, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, ∅

e. {{a}}

{{a}}, ∅

4. Find the power set for each of the following sets

a. {5}

P(a) = {{5}, ∅}

b. {0}

P(b) = {{0}, ∅}

c. {0, 1, 2}

P(c) = {{0}, {1}, {2}, {0,1}, {0, 2}, {1, 2}, {0, 1, 2}, ∅}

d. {{a, b}, c}

P(d) = {{{a, b}}, {c}, {{a, b}, c}, ∅}

e. ∅

P(e) = {∅}

f. P(∅)

P(f) = {∅}

5. Is {1, 3, 5, 7} a subset of {x: x is an odd positive number that is less than 10}, and if so, why?

Yes, because all elements in {1, 3, 5, 7} are members of the other set.



6. Consider if the following statements are true or false.

a. {4} ⊂ {{4}}

False

b. {4} ∈ {{4}}

True

c. ∅ ⊂ {{4}}

True

d. {0} = { }

False

e. ∅ ⊂ { }

True

f. ∅ = 0

False

g. 0 ∈ ∅

False


h. ∅ ∈ ∅

False

i. ∅ ∈ {∅}

True


J. ∅ ⊂ { }
True


7. Consider which of the following sets are subsets of set A

Let A = {x: x is a positive integer that is less than 50}
B = {x: x=2y, y is a positive integer that is less than 11}
C = {x: x=y2, y is an integer from 0 to 5}
D = {x: x is a positive odd number that is less than 50}
B ⊂ A, C ⊄ A, D ⊂ A



8. If A is a set with n numbers, find

a. The number of subsets of A having at least one member
2n-1


b. The number of subsets of A having n-1 members

n number of sets



9. Tell the number of proper subsets of a set with 5 members

2^5-1
31
 
Last edited:
That's a lot to ask us to look through, but most of it is a quick scan. I can't promise not to have missed anything, but the only error I saw was on 3c, which is very subtle. Look closely: the set contains only one element.

Ah, yes -- on a rescan, 4f is also wrong. It might help if rather than write P(f) = ..., you write what it is, P(P(∅)), and then replace P(∅) with what it is, from 4e.

Also, in 8, did you fail to make some things exponents? You may have meant the right things.

On the whole, good work!
 
I didn't mean any inconvenience, but I figured that if someone has a great understanding of set theory, it wouldn't require too much time.

So 3c would just have 2 subsets then, {{1, {3, 5}}} and , correct? That is quite subtle...

4f Would be P(P(∅)), which is the power set of an empty power set, which would be {∅}, correct? so P(P(∅)) = {∅}

for 8a, yes that is my typo, I mean (2^n) - 1

Thank you so much for your help.
 
I didn't mean any inconvenience, but I figured that if someone has a great understanding of set theory, it wouldn't require too much time.

So 3c would just have 2 subsets then, {{1, {3, 5}}} and , correct? That is quite subtle...

4f Would be P(P(∅)), which is the power set of an empty power set, which would be {∅}, correct? so P(P(∅)) = {∅}

for 8a, yes that is my typo, I mean (2^n) - 1

Thank you so much for your help.

You're right -- it didn't take much time, unless I wanted to be really sure not to make a mistake -- which is why I tried to signal to others that they might want to compete with me ...

You've got 3c; think again about 4f. What is P({∅})? Don't forget that {∅} has one element; so how many elements are in its power set? This, too, is subtle -- that's what this whole exercise set is training you to be!
 
So for 4f, {∅} has one element, which means the power set has 2 sets . So the power set of an empty power set would be the set of an empty set, and an empty set? ----> P(P(∅)) = {{∅}, ∅}
 
So for 4f, {∅} has one element, which means the power set has 2 sets . So the power set of an empty power set would be the set of an empty set, and an empty set? ----> P(P(∅)) = {{∅}, ∅}

Correct. Questions like this involving the empty set can easily fool you, so you have to slow down. That's true of me as well! It's still possible that I missed something, but I think you're good.
 
So 4e, which is asking for the power set of ∅, would be {∅} and ∅ as well?
 
So 4e, which is asking for the power set of ∅, would be {∅} and ∅ as well?

No, that is the power set of {∅}.

It may help to rewrite ∅ as {}. Clearly the power set of {} is just {{}}, that is, the only subset is the empty set itself.

The power set of {∅} = {{}} is {{{}},{}}. This is where it's better to use ∅! Or to write
Code:
{ {{}},
  {  } }
That is, it is a set consisting of a set containing the empty set, and an empty set.

 
So for 4f, {∅} has one element, which means the power set has 2 sets . So the power set of an empty power set would be the set of an empty set, and an empty set? ----> P(P(∅)) = {{∅}, ∅}
I would think of {∅}as the set that contains one element and that element is NOT the empty but rather the symbol of the empty set. As long as you think of it that way you will never make any silly mistakes. So for example {∅} does not equal {}/ Why, because the 1st set has an element in it--the element happens to be symbol for the empty set but none the less is an element-- while {} has no elements in it.
 
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