Subset Help Set Theory: Given A = {2, {4, 5}, 4}, consider the following...

[thunc14]

I'm a bit confused on this problem. I know it's simple but the subtle changes and notation are tripping me up.

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

{4, 5} ⊂ A
{4, 5} ∈ A
{{4, 5}} ⊂ A
5 ∈ A
{5} ∈ A
{5} ⊂ A
{2} ⊂ A
{2} ∈ A
2 ⊂ A
2 ∈ A

Thanks in advance for any help.

 

[Dr.Peterson]

I'm a bit confused on this problem. I know it's simple but the subtle changes and notation are tripping me up.

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

{4, 5} ⊂ A
{4, 5} ∈ A
{{4, 5}} ⊂ A
5 ∈ A
{5} ∈ A
{5} ⊂ A
{2} ⊂ A
{2} ∈ A
2 ⊂ A
2 ∈ A

Thanks in advance for any help.

Please show your guesses, so we can know where you need help and focus our attention there. If you can also explain your reasoning, it will be even better.

Example: {4, 5} ⊂ A is false because that requires 4 and 5 to be (individually) elements of A, and 5 is not.

 

[thunc14]

Please show your guesses, so we can know where you need help and focus our attention there. If you can also explain your reasoning, it will be even better.

Example: {4, 5} ⊂ A is false because that requires 4 and 5 to be (individually) elements of A, and 5 is not.

Given A = {2, {4, 5}, 4}, consider if the following statements are true or false
{4, 5} ⊂ A
True, because it is a member of set A, therefore a subset. (I know you said it is false but that was my original guess)

{4, 5} ∈ A
Yes, same reason as above.

{{4, 5}} ⊂ A
I don't know what this means

5 ∈ A
Yes, it is a member of A.

{5} ∈ A
I don't know what this means

{5} ⊂ A
I don't know what this means

{2} ⊂ A
I don't know what this means

{2} ∈ A
I don't know what this means

2 ⊂ A
Yes, it is a member, therefore a possible subset

2 ∈ A
Yes, it is a member

 

[Dr.Peterson]

It will help to think of a set as a bag containing things. Your set A is a bag containing the number 2, the number 4, and another bag containing 4 and 5. The contents of the inner bag are not considered elements of the outer bag, because they are not directly contained.

Given A = {2, {4, 5}, 4}, consider if the following statements are true or false
{4, 5} ⊂ A
True, because it is a member of set A, therefore a subset. (I know you said it is false but that was my original guess)

Member and subset are different things. A member is something that is directly listed (namely 2, {4,5}, of 4. A subset is a set containing some or all of those members. Here, {4,5} is one of the members; its own members, 4 and 5, are not both members of A, so it is not a subset.

{4, 5}

∈ A
Yes, same reason as above.

Your answer is correct, as I stated above. This set is listed as a member.

{{4, 5}}

⊂ A
I don't know what this means

This is a set whose only member is the set {4,5}. Think of a bag containing a bag containing stuff. To see whether it is a subset, you ask, is every member of this set a member of A? The answer is yes, since {4,5} is a member of A.

5

∈ A
Yes, it is a member of A.

Correct.

{5}

∈ A
I don't know what this means

Is {5} listed as a member of A? Again, think of it not as 5 sitting there itself, but 5 contained in a bag sitting inside A. Since {5} is not listed (only 5 itself is), it is not a member.

{5}

⊂ A
I don't know what this means

What do you think now?

{2}

⊂ A
I don't know what this means

What do you think now?

{2}

∈ A
I don't know what this means

What do you think now?

2

⊂ A
Yes, it is a member, therefore a possible subset

No, this is not asking whether 2 is a member, but whether it is a subset. Since 2 is not a set at all, it can't be.

2

∈ A
Yes, it is a member

Correct.

Try again, so I can see whether this has helped.

There are some tricky concepts here; mathematicians have had to develop them in order to talk clearly about sets, though this is foreign to ordinary language! So it does take time to catch on.

 

[thunc14]

Given A = {2, {4, 5}, 4}, consider if the following statements are true or false
{4, 5} ⊂ A
False. {4} ⊂ A is true, but {5} ⊂ A is not true, so the statement is false.

{4, 5} ∈ A
True. The set of 4, 5 is inside the { }, so it is a member of A

{{4, 5}} ⊂ A
True. The set {4, 5} is a member of A, so the set containing this set is a subset of A

5 ∈ A
No, 5 is not directly listed as a member of A

{5} ∈ A
No, there is no set of 5 listed, therefore the statement is false

{5} ⊂ A
No, because the element in {5}, which is 5, is not a direct member of set A.

{2} ⊂ A
Yes, because every element is {2}, which is 2, is a member of set A

{2} ∈ A
No, because there is no {2} listed as a member of set A

2 ⊂ A
No, because this is not asking for a set

2 ∈ A
Yes, 2 is a member of set A

 

[Dr.Peterson]

Given A = {2, {4, 5}, 4}, consider if the following statements are true or false
{4, 5} ⊂ A
False. {4} ⊂ A is true, but {5} ⊂ A is not true, so the statement is false.
I think it would be clearer to say, 4 ∈ A is true, but 5 ∈ A is not true. But you're right.
{4, 5} ∈ A
True. The set of 4, 5 is inside the { }, so it is a member of A

{{4, 5}} ⊂ A
True. The set {4, 5} is a member of A, so the set containing this set is a subset of A

5 ∈ A
No, 5 is not directly listed as a member of A

{5} ∈ A
No, there is no set of 5 listed, therefore the statement is false

{5} ⊂ A
No, because the element in {5}, which is 5, is not a direct member of set A.

{2} ⊂ A
Yes, because every element in {2}, which is 2, is a member of set A

{2} ∈ A
No, because there is no {2} listed as a member of set A

2 ⊂ A
No, because this is not asking for a set

2 ∈ A
Yes, 2 is a member of set A

Good work. Feeling a little less fuzzy yet? It will take time.

 

[Steven G]

I'm a bit confused on this problem. I know it's simple but the subtle changes and notation are tripping me up.

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

{4, 5} ⊂ A
{4, 5} ∈ A
{{4, 5}} ⊂ A
5 ∈ A
{5} ∈ A
{5} ⊂ A
{2} ⊂ A
{2} ∈ A
2 ⊂ A
2 ∈ A

Thanks in advance for any help.

Maybe this will help. The commas between the outer { } separate the set into its elements or members or objects. So the elements in your set are 2, {4,5) and 4. Yes, {4,5} in this case is an element of the set.

Now to use C, you must have a set on both sides of it. So if A and B are sets then we can talk about A C B. So when is A C B, read as A is a subset of B? A C B if every element in A is in B but B has at least one more element than A.

Let A be a set. So 7 C A is always false because 7 is not even a set. Again the only way possible for A to be a subset of B is if A and B are sets. Then you can check the elements of A and B to decide if in fact A C B.