# Write out the following set by listing their elements....

In summary: They also say that {3, 4} is a proper subset of B, but is not an element of B.)I'm not sure why I've been laboring under this misconception. Certainly, if A = {2, 4, 6}, then {2, 4} ∈ A.In summary, a set cannot have repeated elements and the cardinality of a set refers to the number of elements in the set. The empty set is a subset of every set, but it is not considered an element of a set. Therefore, the answer to the given question is simply ∅ as there are no sets that have three elements whose cardinality is four.

## Homework Statement

Write out the following set by listing its elements between braces:

{X : X ⊆ {3,2,a} and |X| = 4}

## The Attempt at a Solution

My book tells me that it is ∅, but couldn't X have repeated elements? so X = {3,2,a,3}, {3,2,a,2} etc for example??

Maybe start out by writing in words what your interpretation of:
{X : X ⊆ {3,2,a} and |X| = 4}

SunThief said:
Maybe start out by writing in words what your interpretation of:

The set X such that X is a subset of set {3,2,a}, and X has carnality (size) of 4 elements

Ok... walk through it. So you know:

1) X is a subset of the given set.
2.) The given set displays 3 elements.
3.) X itself has a size of 4 elements.

- [Then what's missing, and what are the rules for subsets?]​

That's why I'm asking.. can elements be repeated? Would X be able to have... say... the element a, repeating? Wouldn't ∅ have to be in the set {3,2,a} forthe answer to be ∅?

That's why I'm asking.. can elements be repeated? Would X be able to have... say... the element a, repeating? Wouldn't ∅ have to be in the set {3,2,a} forthe answer to be ∅?

No: elements of a set cannot be repeated. Other types of collections can have repeated elements, but not sets.

That's why I'm asking.. can elements be repeated?
No, they can't be repeated.

Also, the empty set is a subset of every set, but it isn't an element of a set.
Your set, {3, 2, a} contains three elements.

One other thing -- the cardinality of a set is the number of elements in it, Carnality means something entirely different.

SunThief and StoneTemplePython
That's why I'm asking.. can elements be repeated? Would X be able to have... say... the element a, repeating? Wouldn't ∅ have to be in the set {3,2,a} forthe answer to be ∅?

Not trying to be coy... just trying to be methodical. You can do a wiki search on "Subset" to verify, but my understanding is that:

- ∅ is a subset of every set.
- repeats aren't allowed.​

Okay thanks for clarifying everyone. That's where my main confusion came from: can sets have repeated elements or not.

Write out the following set by listing its elements between braces:

{X : X ⊆ {3,2,a} and |X| = 4}

isn't simply ∅ like my book says, but instead its {3,2,a,∅}, am I right?

Mark44 said:
One other thing -- the cardinality of a set is the number of elements in it, Carnality means something entirely different.

For some reason spell check doesn't acknowledge that cardinality is a word :s I guess it changed it to carnality on accident.

Okay thanks for clarifying everyone. That's where my main confusion came from: can sets have repeated elements or not.

Write out the following set by listing its elements between braces:

{X : X ⊆ {3,2,a} and |X| = 4}

isn't simply ∅ like my book says, but instead its {3,2,a,∅}, am I right?
No. |X| = 3. There are no sets that have three elements whose cardinality (|X|) is 4. The empty set ∅ is a subset of every set, but it's not an element of a set. In symbols, ∀ A, ∅ ⊂ A, but ∅ ∉ A.

SunThief
Mark44 said:
In symbols, ∀ A, ∅ ⊂ A, but ∅ ∉ A.
It has been eons since I actually took a set theory course, but isn't that statement technically false? What about when ##A = \{ \phi \} ##? Isn't ##\phi## both an element and a subset of ##A##?

Mark44 said:
∀ A, ∅ ⊂ A, but ∅ ∉ A.
LCKurtz said:
It has been eons since I actually took a set theory course, but isn't that statement technically false? What about when ##A = \{ \phi \} ##? Isn't ##\phi## both an element and a subset of ##A##?
It's been eons for me as well, so I'm not dead certain what I wrote is correct. My thinking is that there's a distinction between subsets of a set vs. elements of a set. For example, if A = {2, 4, 6}, then {2, 4} ⊂ A, but we wouldn't write {2, 4} ∈ A.

Further, the cardinality of my set A, or |A| is 3, whilie |∅| is 0. If ∅ were an element of A, wouldn't A's cardinality be 4? (Which gets us back to the question of the OP.)

As a sideline, it turns out that the reason we use ∈ (a stylized ##\epsilon##) is that epsilon is the first letter of the Greek word ἐστί, "is." Transliterated, this would be something like "esti".

LCKurtz said:
It has been eons since I actually took a set theory course, but isn't that statement technically false? What about when ##A = \{ \phi \} ##? Isn't ##\phi## both an element and a subset of ##A##?

Mark44 said:
It's been eons for me as well, so I'm not dead certain what I wrote is correct. My thinking is that there's a distinction between subsets of a set vs. elements of a set. For example, if A = {2, 4, 6}, then {2, 4} ⊂ A, but we wouldn't write {2, 4} ∈ A.

Further, the cardinality of my set A, or |A| is 3, whilie |∅| is 0. If ∅ were an element of A, wouldn't A's cardinality be 4? (Which gets us back to the question of the OP.)
I wasn't referring to the particular A in the OP. Rather, to the dummy variable A in your statement "In symbols, ∀ A, ∅ ⊂ A, but ∅ ∉ A." It seems to me that ##A = \{ \phi \} ## is a counterexample to that statement.

LCKurtz said:
I wasn't referring to the particular A in the OP. Rather, to the dummy variable A in your statement "In symbols, ∀ A, ∅ ⊂ A, but ∅ ∉ A." It seems to me that ##A = \{ \phi \} ## is a counterexample to that statement.
I concede your point. I've been laboring under the misconception that a subset of a set wasn't also considered an element of the set. After searching a number of wiki pages, I found one that says that a subset of a set is also considered an element of the set -- https://en.wikipedia.org/wiki/Element_(mathematics). An example they give is B = {1, 2, {3, 4}}, which they describe as having three elements: 1, 2 and {3, 4}.

## 1. What does it mean to "write out a set by listing its elements?"

When writing out a set, you are essentially creating a list of all the distinct objects or numbers within that set. This means that you are identifying and listing each individual element of the set.

## 2. How do I write out a set?

To write out a set, you can use curly braces { } to enclose the elements of the set, separated by commas. For example, if the set is "even numbers between 1 and 10", you would write it as {2, 4, 6, 8, 10}.

## 3. Can a set have repeating elements?

No, a set cannot have repeating elements. Each element in a set must be unique and cannot be repeated. For example, {1, 2, 2, 3} is not a valid set because the number 2 is repeated.

## 4. What is the difference between a set and a list?

A set is a collection of distinct elements, meaning each element is unique and cannot be repeated. A list, on the other hand, can have repeating elements and the order of the elements is important. Additionally, sets are denoted by curly braces { }, while lists are denoted by square brackets [ ].

## 5. Are sets and subsets the same thing?

No, sets and subsets are not the same thing. A subset is a set that contains elements that are all part of a larger set. For example, {2, 4, 6} is a subset of {1, 2, 3, 4, 5, 6}. However, a subset can also contain all the elements of the larger set, making it equal to the larger set.