Concept

Subset

If all the elements of a set A are contained in another set B, then A is a subset of B. This relationship can be written as A⊆ B. A Venn diagram visualizes the expression.

As an example, consider the set B={1,2,3}. All of its subsets are shown in the table.

Subset With No Elements	{ } or ∅
Subsets With One Element	{ 1 },{ 2 },{ 3 }
Subsets With Two Elements	{ 1,2 },{ 2,3 },{ 1,3 }
Subset With Three Elements	{ 1,2,3 }

It is worth noting that all the subsets of B except { 1,2,3 } are called proper subsets. A proper subset of a set B is a subset with fewer elements than B and is denoted by ⊂. {1,2} ⊂ B Other than that the empty set is a subset of all sets. Additionally, every set is always a subset of itself and it is denoted by ⊆. This fact is illustrated with set B. { } ⊂ B { 1,2,3 } ⊆ B

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Another Example

Actually, sets and subsets do not necessarily include just the numbers all the time. Remember that any well-defined object can be an element of a set. For instance, consider the following sets written in verbal description. Set A:&Water bodies of the world Set B:&Oceans Set C:&Lakes Since all the oceans and lakes are also water bodies of the world, both the sets B and C are the subsets of set A.

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