sets

A set is a collection of elements, often numbers, which can be indicated by listing these elements in braces {}.

For example,

The set of whole numbers less than 7 is {1, 2, 3, 4, 5, 6}

Some sets have an infinite number of elements, such as the set of all integers greater than -2: {-1, 0, 1, 2, 3,…}

• A set containing no elements is called the empty set, or null set, and is indicated by the symbol Ø or { }.

• The set that contains all objects is

called the universal set or the universe, usually labeled U.

• Any set is a subset of the universe.

• If a set is called A, then the complement of A, written A’, is every element in the universe U that is not in A.

• The intersection of two sets is the set containing all the elements in common to the two sets. An intersection of sets is denoted by the symbol between the set names.

• Disjoint sets are sets that have no elements in common, so their intersection is the empty set.

• The union of two sets is the set of elements in one set or the other set. The symbol for this operation is .

Exercises1)Given A = {3, 4, 5, 6} andB = {3, 6, 9, 12},find A B and A B.

2) Let C = {-4, -2, 0, 2} and

D = {1, 2, 3}, be two sets in universe

U = {-4, -3, -2, -1, 0, 1, 2, 3, 4}.

Find C’ and D’.

• The cross product or Cartesian product of two sets A and B, denoted as A x B, is the set of all ordered pairs (a, b) where a is an element of A and b is an element of B.

Note that B x A is not the same as A x B and will not be equal to it unless A = B or one of the two sets is the empty set.

Given that A = {3, 4, 5} and

B = {1, 2, 3},

find A x B and B x A.

A Venn Diagram is useful for sorting and classifying sets of objects or numbers. Each circle in a Venn diagram represents a subset of objects within the universal set (represented by the rectangle). The overlapping regions indicate objects that the subset have in common. An object that does not belong to a subset appears outside the circles.

Example 1The Venn diagram shows how many

students in a tenth grade class are taking Spanish, French, and Mandarin.

• How many students are studying Spanish?

• How many are studying only French?

• How many are studying both Spanish and Mandarin?

• How many are studying Spanish, French and Mandarin?

Example 2In the Venn diagram at the right, the

universal set is {1, 2, 3, …9}• What is the intersection of sets A

and B? • What is the intersection of sets A,

B, and C? • What is the intersection of sets A

and C? • What is the union of sets B and C?

Example 3Sets A and B are both in

universe U but are not equal. Therefore, they are either Disjoint Sets (having no elements in common) or overlapping sets. Neither is a subset of the other.