Set operations are fundamental to understanding the relationships between collections of objects, whether numbers, characters, or other entities. These operations enable us to combine, intersect, and differentiate sets in meaningful ways. This guide offers a concise overview of set operations, employing set notation and examples to elucidate concepts.

Introduction to Set Operations

Sets are collections of distinct objects, termed elements. They can be represented in various forms, including statement, roster, and set builder notations. Set operations, including the union, intersection, complement, and difference of sets, help establish relationships between sets, akin to arithmetic operations with numbers. Venn diagrams are invaluable for visualising these relationships.

Key Set Operations

Union of Sets $(A∪B)$

The union of sets A and B, denoted as A∪B, comprises all distinct elements from both sets. The formula $n(A∪B) = n(A) + n(B) − n(A∩B)$ calculates the total number of elements in the union, where n(X) represents the number of elements in set X.

Example:

If A = {1, 2, 3, 4} and B = {4, 5, 6, 7}, then $A ∪ B$ = {1, 2, 3, 4, 5, 6, 7}.

Intersection of Sets $(A∩B)$

The intersection of sets A and B, denoted A∩B, includes all elements common to both sets.

Example:

If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, $A ∩ B$ = {3, 4}.

Set Difference $(A - B)$

The difference between sets A and B, expressed as A - B, lists elements in A not found in B.

Example:

If A = {1, 2, 3, 4} and B = {3, 4, 5, 7}, $A - B$ = {1, 2}.

Complement of a Set (A′)

The complement of set A, noted as A′ or Ac, consists of all elements in the universal set U not in A.

Example:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, then A′ = {5, 6, 7, 8, 9}.

Properties of Set Operations

Set operations adhere to several properties analogous to arithmetic operations:

• Commutative Law: $A ∪ B = B ∪ A and A ∩ B = B ∩ A$
• Associative Law: $(A ∪ B) ∪ C = A ∪ (B ∪ C)$ and $(A ∩ B) ∩ C = A ∩ (B ∩ C)$
• De Morgan's Law: $(A ∪ B)' = A' ∩ B'$ and $(A ∩ B)' = A' ∪ B'$

Other notable properties include the idempotent, identity, and subset properties.

Important Notes

• The formula for the union of sets is $n(A∪B) = n(A) + n(B) − n(A∩B)$, and for the intersection of sets is $n(A∩B) = n(A) + n(B) − n(A∪B).$
• Union with the universal set results in the universal set, while intersecting any set with the universal set yields the original set.
• The complement of the universal set is the empty set (U′ = ϕ), and the complement of an empty set is the universal set (ϕ′ = U).