1. 5.1 Sets Sets and Elements Union and Intersection Subset and Empty Set
Universal Set and Complement

2. Sets and Elements
A set is any collection of objects. The objects, which may be countries, cities, years, numbers, letters, or anything else, are called the elements of the set. A set is often specified by a listing of its elements inside a pair of braces. A set may also be specified by giving a description of its elements.

3. Example Sets and Elements
The set of the first six letters of the alphabet is {a, b, c, d, e, f}.
{2, 4, 6, 8, 10} is the set {the even numbers between 1 and 11}.
The graph {(a,b) where b = a2} is a set with infinitely many elements.

4. Example Sets and Elements (2)
Let A = {years from 1991 to 2004 in which unemployment is at least 6%}.
Let B = {years from 1991 to 2004 in which inflation is at least 3%}.

5. Example Sets and Elements (3)
Using the table, the two sets are
A = {1991, 1992, 1993, 1994, 2003} and
B = {1991, 1992, 1993, 1996, 2000}.

6. Union and Intersection
The union of A and B, denoted A B, is the set consisting of all those elements that belong to either A or B or both. The intersection of A and B, denoted A B, is the set consisting of those elements that belong to both A and B.

7. Example Union and Intersection
Let A = {1991, 1992, 1993, 1994, 2003} and
B = {1991, 1992, 1993, 1996, 2000} from the previous example. Find
A B
A B
= {1991,1992,1993,1994,1996,2000,2003}
= {1991,1992,1993}

8. Subset and Empty Set
A set B is called a subset of A provided that every element of B is also an element of A.
The set that contains no elements at all is the empty set (or null set) and is written as . The empty set is a subset of every set.

9. Example Subset and Empty Set
List all possible subsets of {a, b, c}.
{a}, {b}, and {c}
{a, b}, {a, c}, and {b, c}
{a, b, c}

10. Universal Set and Complement
The set U that contains all the elements of the sets being discussed is called a universal set (for the particular problem).
If A is a subset of U, the set of elements in U that are not in A is called the complement of A, denoted by A'.

11. Example Universal Set and Complement
Let U = {a,b,c,d,e,f,g}, S = {a,b,c} and T = {a,c,d}.
Find
= {d,e,f,g}
= {b,e,f,g}
= {b,d,e,f,g}

12. Summary Section Part 1
A set is a collection of objects. Each object is called an element of the set. The empty set is the set containing no objects.
The union of two sets is the set consisting of all elements that belong to at least one of the sets. The intersection of two sets is the set consisting of all elements that belong to both of the sets.

13. Summary Section Part 2
Set A is a subset of set B if every element of set A is also an element of set B. In each situation or problem, all sets are considered to be subsets of a universal set. The set of all elements in the universal set that do not belong to the set A is called the complement of A, denoted A'.