Subset

Definition

A set is a subset of another set if every element of the first set is also in the second set.

Example:

A = {1, 2}
B = {1, 2, 3}

A is a subset of B because every element in A is also in B.

Subset = every element is included in the larger set.

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Subset vs Element

An element is an object inside a set.

A subset is a set inside another set by membership of its elements.

Example:

B = {1, 2, 3}

Element:

2

Subset:

{1, 2}

The element 2 is not the same type of thing as the subset {1, 2}.

Element belongs to a set; subset is contained by element inclusion.

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Every Set Is A Subset Of Itself

Example:

A = {1, 2, 3}

Every element of A is in A.

So:

A is a subset of A

This may feel obvious, but it matters for precise reasoning.

A set is always a subset of itself.

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Proper Subset

A proper subset is contained in another set but is not equal to it.

Example:

A = {1, 2}
B = {1, 2, 3}

A is a proper subset of B.

Why?

Because B has at least one element not in A.

Proper subset means contained but not equal.

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Final Summary

• A subset is a set whose elements all belong to another set.
• Subset is different from element.
• Every set is a subset of itself.
• A proper subset is contained in another set but not equal to it.
• Subsets let us reason about containment between collections.