The Basic Concepts of Set Theory

Chapter 1 Set Operations and Cartesian Products

Intersection of Sets The intersection of sets A and B, written is the set of elements common to both A and B, or

Find each intersection. a) b) Solution a) b) Example: Intersection of Sets

Union of Sets The union of sets A and B, written is the set of elements belonging to either of the sets, or

Find each union. a) b) Solution a) b) Example: Union of Sets

Difference of Sets The difference of sets A and B, written A – B, is the set of elements belonging to set A and not to set B, or

Example: Difference of Sets Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, h}, B = {c, e, g}, and C = {a, c, d, g, e}. Find each set. a) b) Solution a) {a, b, h} b)

Universal Set and Subsets The Universal Set denoted by U is the set of all possible elements used in a problem. When every element of one set is also an element of another set, we say the first set is a subset. Example A={1, 2, 3, 4, 5} and B={2, 3} We say that B is a subset of A. The notation we use is B A. Let S={1,2,3}, list all the subsets of S. The subsets of S are, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.

The Empty Set The empty set is a special set. It contains no elements. It is usually denoted as { } or. The empty set is always considered a subset of any set. Is this set {0} empty? It is not empty! It contains the element zero.

Cardinal Number The Cardinal Number of a set is the number of elements in the set and is denoted by n(A). Let A={2,4,6,8,10}, then n(A)=5. The Cardinal Number formula for the union of two sets is n(A U B)=n(A) + n(B) – n(A∩B). The Cardinal number formula for the complement of a set is n(A) + n(A ’ )=n(U).

Ordered Pairs In the ordered pair (a, b), a is called the first component and b is called the second component. In general Two ordered pairs are equal provided that their first components are equal and their second components are equal.

Cartesian Product of Sets The Cartesian product of sets A and B, written, is

Example: Finding Cartesian Products Let A = {a, b}, B = {1, 2, 3} Find each set. a) b) Solution a) {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)} b) {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Cardinal Number of a Cartesian Product If n(A) = a and n(B) = b, then

Example: Finding Cardinal Numbers of Cartesian Products If n(A) = 12 and n(B) = 7, then find Solution

Venn Diagrams of Set Operations A B U AB U A U AB U

Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set AB U

Example: Shading Venn Diagrams to Represent Sets Draw a Venn Diagram to represent the set A B C U