Basic Set Operations: Union, Intersection, and Complement

Learn fundamental set operations: union, intersection, and complement. This tutorial provides clear definitions, illustrative examples using Venn diagrams, and explains how these operations combine and manipulate sets, forming a foundation for understanding set theory and its applications.

Basic Set Operations

Sets are collections of unique items. Several operations allow us to combine and manipulate sets.

1. Union of Sets (∪)

The union of two sets A and B (A ∪ B) is the set containing all elements that are in A, in B, or in both. Think of it as an "or" operation.

Example: Union of Sets

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

2. Intersection of Sets (∩)

The intersection of two sets A and B (A ∩ B) is the set containing only the elements that are in both A and B. Think of it as an "and" operation.

Example: Intersection of Sets

If A = {11, 12, 13} and B = {13, 14, 15}, then A ∩ B = {13}.

3. Difference of Sets (-)

The difference between two sets A and B (A - B) is the set of elements that are in A but not in B.

Example: Difference of Sets

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

4. Complement of a Set (c)

The complement of a set A (Ac) contains all the elements in the universal set (U) that are not in A. The universal set contains all the elements relevant to a given context.

Example: Complement of a Set

If U is the set of all natural numbers and A = {1, 2, 3}, then Ac is the set of all natural numbers except 1, 2, and 3.

5. Symmetric Difference of Sets (⊕)

The symmetric difference of sets A and B (A ⊕ B) contains all elements that are in A or B, but not in both. It's like the union minus the intersection.

Example: Symmetric Difference of Sets

If A = {a, b, c, d} and B = {a, b, l, m}, then A ⊕ B = {c, d, l, m}.

Conclusion

These five operations—union, intersection, difference, complement, and symmetric difference—provide the fundamental tools for manipulating and analyzing sets in discrete mathematics.