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

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Homework Statement


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

{X : X ⊆ {3,2,a} and |X| = 4}

Homework Equations




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??
 

Answers and Replies

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


... is actually asking.
The set X such that X is a subset of set {3,2,a}, and X has carnality (size) of 4 elements
 
  • #4
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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?]​
 
  • #5
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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 ∅?
 
  • #6
Ray Vickson
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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.
 
  • #7
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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.
 
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  • #8
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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.​
 
  • #9
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Okay thanks for clarifying everyone. That's where my main confusion came from: can sets have repeated elements or not.

So the answer to:
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?

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.
 
  • #10
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Okay thanks for clarifying everyone. That's where my main confusion came from: can sets have repeated elements or not.

So the answer to:
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.
 
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  • #11
LCKurtz
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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##?
 
  • #12
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Mark44 said:
∀ 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##?
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".
 
  • #13
LCKurtz
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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.)
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.
 
  • #14
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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}.
 

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