Set Elements: 2^\emptyset and {x:x \subseteq {1,2,3,4,5} and |x| \leq 1}

In summary, the power set of the empty set is the set containing only the empty set, and the set S contains only the empty set as an element.
  • #1
cragar
2,552
3

Homework Statement


Write out the following sets by listing their elements between the curly braces.

a. [itex] 2^\emptyset [/itex]
b. {x:x [itex]\subseteq[/itex] {1,2,3,4,5} and |x|[itex] \leq [/itex] 1

The Attempt at a Solution


a. I'm not sure if the only element is one, or if this is an undefined operation.
b. I think the elements of this set are [itex] {\emptyset} [/itex] and {1}
 
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  • #2
I would say
a) {1}
b) {1}
 
  • #3
for part b. I think the empty set should be included because it says that x is a subset of that set and the empty set is a subset of any set.
 
  • #4
cragar said:
a. I'm not sure if the only element is one, or if this is an undefined operation.
Given any set S, we can always form its power set. This is just the set of all subsets of S. What are the subsets of the empty set?

b. I think the elements of this set are [itex] {\emptyset} [/itex] and {1}

I think you're misunderstanding the question. |S|≤ 1 means that the cardinality of S is less than or equal to 1, i.e., S has one element or fewer. What subsets can you form with one element or fewer?
 
  • #5
ok so the only subset of the empty set is the empty set so it has 1 element.
and then my answer for part b will just have 1 element.
 
  • #6
cragar said:
ok so the only subset of the empty set is the empty set so it has 1 element.

Right. So [itex] \mathcal{P}(\emptyset) = [/itex] ?

and then my answer for part b will just have 1 element.

I think you're getting a little mixed up. Remember that elements of the power set are in fact sets themselves!

Let's call [itex] S = \{X : X \subseteq \{1, 2, 3, 4, 5\}[/itex] and [itex] |X| \leq 1\}[/itex]. I claim that the set {2} is in S. Do you see why?
 
  • #7
spamiam said:
Right. So [itex] \mathcal{P}(\emptyset) = [/itex] ?



I think you're getting a little mixed up. Remember that elements of the power set are in fact sets themselves!

Let's call [itex] S = \{X : X \subseteq \{1, 2, 3, 4, 5\}[/itex] and [itex] |X| \leq 1\}[/itex]. I claim that the set {2} is in S. Do you see why?
Thanks for your help.

So the power set of the empty set is the empty set.
and {2} is in S because it has just 1 element.
 
  • #8
cragar said:
Thanks for your help.

So the power set of the empty set is the empty set.

Not quite. The only element of [itex] \mathcal{P}(\emptyset)[/itex] is [itex]\emptyset[/itex]. So [itex] \mathcal{P}(\emptyset)[/itex] can't be empty: I just named one of its elements. Things get tricky when we have sets whose elements are sets whose elements are sets...

So, [itex] \mathcal{P}(\emptyset) [/itex]=?
and {2} is in S because it has just 1 element.

Right, so can you write out S explicitly, listing all its members?
 
  • #9
so the power set of the empty set has 1 element.
 
  • #10
cragar said:
so the power set of the empty set has 1 element.

Yes, there is one subset of the empty set. The empty set.
 

Related to Set Elements: 2^\emptyset and {x:x \subseteq {1,2,3,4,5} and |x| \leq 1}

1. What is the cardinality of the set 2?

The cardinality of a set is the number of elements in the set. In the case of 2, the set is empty and therefore has no elements. This means the cardinality of 2 is 0.

2. How many subsets can be formed from the set {1,2,3,4,5} with a cardinality of 1 or less?

There are a total of 6 subsets that can be formed from the set {1,2,3,4,5} with a cardinality of 1 or less. These subsets are: ∅, {1}, {2}, {3}, {4}, and {5}. This is because a subset with a cardinality of 1 can only contain one element from the original set, and there are 5 possible elements to choose from.

3. Is the set 2 equal to the power set of {1,2,3,4,5} with a cardinality of 1 or less?

Yes, the set 2 is equal to the power set of {1,2,3,4,5} with a cardinality of 1 or less. This is because 2 is the set of all possible subsets of ∅, which is only the empty set itself. And the power set of {1,2,3,4,5} with a cardinality of 1 or less contains all subsets of the original set with a cardinality of 1 or less, including the empty set.

4. Can the set {x:x ⊆ {1,2,3,4,5} and |x| ≤ 1} be simplified further?

No, the set {x:x ⊆ {1,2,3,4,5} and |x| ≤ 1} is already in its simplest form. This set is simply a collection of subsets of {1,2,3,4,5} with a cardinality of 1 or less. It cannot be simplified further without losing information about the elements in the set.

5. What is an example of an element in the set {x:x ⊆ {1,2,3,4,5} and |x| ≤ 1}?

An example of an element in this set is the subset {3}, which contains only the element 3 from the original set {1,2,3,4,5}. This subset has a cardinality of 1 and is a subset of {1,2,3,4,5}, making it an element of the set {x:x ⊆ {1,2,3,4,5} and |x| ≤ 1}.

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