Set contains elements that are *themselves* sets

In summary, a set is a collection of distinct elements grouped together, which can include numbers, letters, or other sets. Sets can contain other sets as elements, known as nested sets or subsets. This does not fundamentally change the principles of set theory. However, a set cannot contain itself as an element, as this leads to logical contradictions. Sets containing sets are commonly used in mathematics to represent relationships between objects or concepts, allowing for more complex and efficient calculations.
  • #1
cepheid
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If a set contains elements that are *themselves* sets, then would it be correct to say that one of those elements is a subset of the set?

E.g.

Q = { {a}, {b}, {c} }

I know that { {a} } is a subset of Q (because it contains ONE of the elements)

But is {a} a subset of Q?

I don't think so, because none of the elements of {a} are present in Q.
 
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  • #2
cepheid said:
I don't think so, because none of the elements of {a} are present in Q.
You are correct.
 
  • #3
thanks
 

FAQ: Set contains elements that are *themselves* sets

What is a set?

A set is a collection of distinct elements, also known as members, that are grouped together. The elements in a set can be anything, such as numbers, letters, or other sets.

How can a set contain elements that are themselves sets?

A set can contain any type of element, including other sets. This is known as a nested set or a subset. For example, the set {1, {2, 3}} contains two elements, the number 1 and the set {2, 3}.

How is a set with sets as elements different from a normal set?

A set with sets as elements is not fundamentally different from a normal set. It just means that the elements themselves happen to be sets. The same principles of set theory still apply, such as the concept of a universal set and the operations of union, intersection, and complement.

Can a set contain itself as an element?

No, a set cannot contain itself as an element. This is known as the Russel's paradox and it leads to logical contradictions. For example, if the set A contains itself, then it would also contain all of its elements, which includes itself, leading to an infinite loop.

What is the significance of sets containing sets in mathematics?

Sets containing sets are commonly used in mathematics to represent relationships between different objects or concepts. For example, in graph theory, a set of vertices can be represented as a set of sets, with each set containing the adjacent vertices to a particular vertex. This allows for more complex and efficient mathematical representations and calculations.

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