The Language of Sets Set theory

1. The Language of SetsSet theory Chapter 2 Sec. 1

2. Key Words • What is a set? • Collection of objects. • Use of capital letters to name sets. • What is a element/member? • Individual objects in a set. • Use of lowercase letters to denote elements in a set.

3. How to represent a set? • Consider the set of seasons of the year to be the set S. • S = {Spring, Summer, Fall, Winter}.

4. Set Builder Notation • Set builder notation is to represent the set, if the elements of a set all share some common characteristics that are satisfied by no other objects.

5. Examples • C = {x:x is a carnivorous animal} • is is equivalent to = • { is the set • x “of all x” • : is such that

6. We can use set builder notation for the solution we will have to write. • C={lion, tiger, panther}

7. Write an alternative method. • B={y:y is a color of the state of New Mexico flag.} • B={yellow and red} • A={a:a is counting number less than 20 and is evenly divisible by 3.} • A={3,6,9,12,15,18}

8. Well defined • A set is well defined if we are able to tell whether any particular object is an element of the set.

9. Example • Here is two examples, which sets are well defined? • A) M = {x:x is a mountain over 10,000 ft high} • Well defined • B) S={s:s is a scary movie} • Not well defined

10. How about this problem? • M = {m:m is in your math class and is also a star on the Sopranos.} • This set has no elements.

11. Empty set or Null set • The set that contains no elements is called the empty set. This set is labeled by the symbol Ø. Another notation for the empty set is {}.

12. Universal set • Is the set of all elements under consideration in a given discussion. We often denote the universal set by the capital U.

13. Example • Consider U = {0, 1, 2, 3, …9, 10} • U = {x:x is a male consumer living in the United States.}

14. Elemental symbol • We will use the symbol to stand for the phrase is an element of. • How is it used?

15. Example • The notation is expresses that 4 is an element of the set A. • The notation is expresses that 4 is not an element of the set A.

16. Use either • A) 3 {2, 4, 3, 5} • B) {4} {2, 3, 4, 5} • 4 {x:x is an odd counting number}

17. Cardinal Number • The number of elements in set A and denoted by n(A). A set is finite if its cardinal number is a whole number. An infinite set is one that is not finite.

18. Example problems • State whether the set is finite or infinite. If it is finite, state its cardinal number using n(A) notation. • P = {x:x is a planet in our solar system}. • N ={1,2, 3}