Understanding Subsets and Elements: Examining the Truth of ∅∈{0}

In summary, the null set is not an element of a set, and subsets of a set must include the set itself.
  • #1
Bashyboy
1,421
5

Homework Statement


The question asks me to determine whether the statement is true or false, the statement being .∅∈{0}


Homework Equations





The Attempt at a Solution


I said that the statement was true, but apparently it is false. Wouldn't a set such as {1,{1}} be made up of the elements 1 and {1}, having a set as one of its elements? So the set in the problem could be written as, {0, ∅}, meaning that the null set is a subset, and also an element?
 
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  • #2
Bashyboy said:

Homework Statement


The question asks me to determine whether the statement is true or false, the statement being .∅∈{0}


Homework Equations





The Attempt at a Solution


I said that the statement was true, but apparently it is false. Wouldn't a set such as {1,{1}} be made up of the elements 1 and {1}, having a set as one of its elements? So the set in the problem could be written as, {0, ∅}, meaning that the null set is a subset, and also an element?

{0} is a set containing a single element. You have ##0 \in \{0\}## but that is the only element. The null set is not a member of this set. The only subsets of {0} are itself and the empty set ∅. It would be correct to write ##\Phi \subset \{0\}##.
 
  • #3
How about {1, {1}}, what are the elements of this set? 1 and {1}?
 
  • #4
Bashyboy said:
How about {1, {1}}, what are the elements of this set? 1 and {1}?

This is true ^. Also, the empty set is a proper subset in regards to your question.
 
  • #5
Okay, so then sets can also be elements, except for the null set?
 
  • #6
The null set can be an element of a set. Example: {1,{1},∅}. Now the null set is both an element and a subset of that set.
 
  • #7
So, then the book's answer to the question in the original post was wrong? Or does the null set have to be explicitly written, like you, Dansuer, did?
 
  • #8
Yes, it needs to be explicitly written. {0} ≠ {0, ∅}. ∅ is an element of the second set but not of the first. While ∅ is a subset of both sets.
If something is a subset of a set, it does not mean it is an element of that set.

For example this is wrong:
{1,2,3} is a subset of {1,2,3,4} and so we write {1,2,3,4,{1,2,3}} and we say that {1,2,3} is an element of the set. This is obliviously wrong, but when we are dealing with the null set it's more confusing.
We usually represent a set with {} while we represent the empty set with just ∅. So ∅ might appear to be an element, while actually it a set.
 
  • #9
Oh, okay, I understand. Thank you.
 

1. What is a subset and how is it different from an element?

A subset is a collection of elements from a larger set. It is different from an element because an element is a single item within a set, while a subset can contain multiple elements from the same or different sets.

2. How do you determine if a set is a subset of another set?

To determine if a set is a subset of another set, you must check if all the elements in the first set are also present in the second set. If so, then the first set is a subset of the second set.

3. Can a set be a subset of itself?

Yes, a set can be a subset of itself. This is because all the elements in the set are also present in itself, satisfying the definition of a subset.

4. What is the difference between a proper subset and an improper subset?

A proper subset is a subset that is not equal to the original set, meaning it does not contain all the elements of the original set. An improper subset is a subset that is equal to the original set, meaning it contains all the elements of the original set.

5. How do you represent subsets and elements in set notation?

In set notation, a subset is represented by the symbol ⊆ and an element is represented by the symbol ∈. For example, if set A is a subset of set B, it would be written as A ⊆ B. If an element x is in set A, it would be written as x ∈ A.

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