Subsets

Subsets

Subsets are sort of like nested Russian dolls:the subset “fits inside” the set.

Subsets

Set A is a subset of set B if all of the members in set A are also in set B.

set A = { 1, 2, 3 } set B = { 1, 2, 3, 4, 5, 6 }

Set A is a subset of set B A B

NOT a Subset

Set A is NOT a subset of set B if any member in set A is not also in set B.

set A = { 1, 2, 9 } set B = { 1, 2, 3, 4, 5, 6 }

Set A is NOT a subset of set B because of the 9.

A B

Proper Subsets

If set A has fewer elements than set B, it is called a proper subset and we use

The idea is very similar to the concept of “less than”

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Improper Subsets

If they are the same set, it is called an improper subset and we use (or we could use =).

By definition, every set is a subset of itself.

A A

Generic Subsets

If we aren’t sure if it’s a proper subset or an improper subset, we use the generic subset symbol.

That way it is true either if it is proper or improper.

The Empty Set

Also by definition:

The empty set is a subset of every other set.

For every set A:

A

Listing Subsets

Let’s list all the subsets of the set {1, 2}

The proper subsets: {1}, {2}, and The improper subset: {1, 2}

A set with 2 elements has 22 or 4 subsets

A set with 2 elements has 22 – 1 or 3 proper

subsets

Listing Subsets

Let’s list all the subsets of the set {1, 2, 3}

The proper subsets: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}The improper subset: {1, 2, 3}

A set with 3 elements has 23 or 8 subsets

A set with 3 elements has 23 – 1 or 7 subsets

Number of Subsets

If a set has n elements

The number of subsets will be 2n

The number of proper subsets will be 2n – 1

Vocabulary

Subset

Proper subset

Improper subset

Empty set