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Q: Does all sets have subsets

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If a set has N elements then it has 2N subsets. So you can see that a list of all subsets soon becomes a very big task. For reasonably small values of N, one way to generate all subsets is to list the binary numbers from 0 to 2N. Then, each of these represents a subset of the original set. If the nth digit is 0 then the nth element is not in the set and if the nth digit is 1 then the nth element is in the set. That will generate all the subsets.

No. The number of subsets of that set is strictly greater than the cardinality of that set, by Cantor's theorem. Moreover, it's consistent with ZFC that there are two sets which have different cardinality, yet have the same number of subsets.

Partitioning is dividing a set of things into subsets such that the union of all the subsets is the original set and the intersection of any two subsets is the null set. That is, between them, the subsets account for the whole of the original set and there are no elements in more than one subset.

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The subsets of all the square roots of 144 are {+12} and {-12}. The single set that includes all the square roots of 144 is {+12, -12}. That's all there are. There are no more.

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Subsets of Sets was created in 2001-08.

Elements can belong to subsets. Subsets can be elements of sets that are called "power sets".

An element doesn't have subsets. Sets can have subsets.

For example, if you take the set A = {1, 2}, then the following sets are all subsets of it: {}, {1}, {2}, {1, 2}. That is, all the sets that fulfill the condition that all of its elements are also elements of the set "A".

Integers, rationals. Also all subsets of these sets eg all even numbers, all integers divided by 3.

Subsets is one type of an sets

Venn diagrams are a means for displaying relationships between sets.

If the set is of finite order, that is, it has a finite number of elements, n, then the number of subsets is 2n.

If a set has N elements then it has 2N subsets. So you can see that a list of all subsets soon becomes a very big task. For reasonably small values of N, one way to generate all subsets is to list the binary numbers from 0 to 2N. Then, each of these represents a subset of the original set. If the nth digit is 0 then the nth element is not in the set and if the nth digit is 1 then the nth element is in the set. That will generate all the subsets.

If the set has "n" elements, then you can make 2n different subsets. The number of subsets will always be greater than the size of the set, both for finite and for infinite sets.

If tiu have a set S, its power set is the set of all subsets of S (including the null set and itself).

Any names that you like. Often they are denoted by CAPITAL letters.

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