Sets: Membership, Subsets, Union, Intersection and Difference

Written by Jean Carlos Novaes

A set is a collection whose members can be identified unambiguously. Listing its elements in braces, describing a shared property or drawing a Venn diagram are different representations of the same idea. Order and repeated listings do not create new elements.

Membership compares an element with a set

The membership symbol asks whether an object occurs in a set. Inclusion instead asks whether every element of one set belongs to another. A set can itself be an element of a larger set, so inspect the braces carefully.

1∈{1,2,3}
{1}⊆{1,2,3}
{1}∈{{1},2,3}

These three statements are all valid but mean different things. The number 1 and the singleton containing 1 are different objects. Every set is a subset of itself, and the empty set is a subset of every set. A singleton has one element; the empty set has none.

Worked operations on two finite sets

Take A and B below. Union collects elements from either set, counting shared members once. Intersection keeps only common elements. Difference keeps elements of the first set that do not belong to the second.

A={1,2,3,4}
B={1,4,6}
A∪B={1,2,3,4,6}
A∩B={1,4}
A∖B={2,3}
B∖A={6}
The overlap contains 1 and 4, common to A and B. The left-only region contains 2 and 3; the right-only region contains 6.
The overlap contains 1 and 4, common to A and B. The left-only region contains 2 and 3; the right-only region contains 6.

The two differences are not equal, so subtraction of sets is not commutative. Union and intersection are commutative and associative. The empty set leaves a union unchanged and makes an intersection empty.

A∪∅=A
A∩∅=∅
A∪B=B∪A
A∩B=B∩A

Complement depends on the universe

Choose a universe U containing the objects under discussion. The complement of A contains elements of U that are outside A. Changing U can change the complement even when A stays fixed. De Morgan’s laws relate complements to union and intersection using the same universe throughout.

(A∪B)c=Ac∩Bc
(A∩B)c=Ac∪Bc

The power set contains subsets, not just original elements

For S containing 1, 2 and 3, its power set has the empty set, the three singletons, the three two-element subsets and S itself. There are eight subsets because each of the three elements can independently be included or excluded.

S={1,2,3}
|P(S)|=23=8

Two sets are equal exactly when each is a subset of the other. If your course uses the strict-subset symbol, equality must be excluded; the subset-or-equal symbol avoids ambiguity. Apply these distinctions to number sets and intervals.

References