1. Copyright © 2014 Curt Hill Sets Introduction to Set Theory

2. Introduction Fundamental discrete structure A set is a collection of distinct items A set has no order No duplications An item is in the set or not –Just as a proposition has two truth values Set variables are usually denoted by capital letters and the items by lower case Copyright © 2014 Curt Hill

3. Terminolgy Set: –A collection of distinct items. –Set variable is usually a capital letter. –Braces contain the elements Element: aka member –One of the items in a set. –Usually denoted by lower case letters. –Symbol x  A, x is member of A. Copyright © 2014 Curt Hill

4. Terminology 2 Empty set: –A set with zero members –Symbol is  or { } Disjoint set: –Two sets with no members in common Cardinality: –The number of elements in a set Universe of discourse: –The set of all those elements that under consideration –Often the integers or real numbers. Copyright © 2014 Curt Hill

5. Subset: A set whose members are all contained in another set The empty set is the subset of every set Opposite of superset A proper subset has at least one element that is present in the superset and not present in the subset An improper subset is the set itself Copyright © 2014 Curt Hill

6. Notation A is a (proper) subset of B (A  B) A is a (proper or improper) subset of B (A  B) A is proper superset of B (A  B) A is superset of B (A  B) Copyright © 2014 Curt Hill

7. Defining a Set There are typically three ways to define a set Enumeration Set builder Construction using operators Copyright © 2014 Curt Hill

8. Enumeration Lists each element in the set –A={1,2,3,4,5} AKA Roster method May use an ellipsis to show a large or infinite set –A={2,4,6,8,…} –A={2,4,6,8,…98,100} Copyright © 2014 Curt Hill

9. Set Builder Notation Uses a rule that defines the members that are present in the set –{x|x  I and x>0 and x<5} or {x|x  I and 0 < x <5} –The | is read such that –I is the set of integers The expression to the right should give a Boolean value as to whether this is a member or not Copyright © 2014 Curt Hill

10. Open and Closed If the type of number is left out, reals should be assumed –{x|0 < x < 5} We cannot say which is the highest and lowest element of this set –We term of this is open –Interval notation is (0,5) However the following is closed –{x|0  x  5} –Interval notation is [0,5] Copyright © 2014 Curt Hill

11. Construction The third way is to define a set in terms of others sets using set operations, eg union, intersection, etc We will see this as we investigate the operators –Section 1.2 and a different presentation Copyright © 2014 Curt Hill

12. Power Sets A power set is the set of all subsets –Useful for testing all combinations of subsets Consider A = { 1, 2, 3} The power set would then be: P(A) = { , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } Copyright © 2014 Curt Hill

13. Tuples The order of sets is irrelevant –{1, 2, 3} = {3, 1, 2} = {2, 1, 3} In many cases we create ordered tuples For example, we use Cartesian Coordinates to indicate a point in two space Here order is important –(2,3) is not the same point as (3,2) –This is an ordered pair Three space would use an ordered triple Copyright © 2014 Curt Hill

14. Cartesian Product A Cartesian Product creates a set of ordered pairs Denoted by A ⨯ B The resulting set of ordered pairs has all possible combinations where the first element is from A and the second from B Copyright © 2014 Curt Hill

15. Example Suppose: –A = {1, 2, 3} –B = {x, y} Then A B = { {1,x}, {1,y}, {2,x}, {2,y}, {3,x}, {3,y} } Notice that the two sets do not need the same type of elements This can be extended to create n- tuples of any size Copyright © 2014 Curt Hill

16. Connections In the previous chapter we used the “is an element of” symbol  to show the domain of quantified expressions –  x (P(x)  x    x>0) This is re-introduced in 2.1 with an addition –  x   (x>0) (P(x)) The first part restricts the domain to integers greater than zero Copyright © 2014 Curt Hill

17. Truth Sets Rosen defines a truth set in a way similar to a solution set from the Algebra of Real Numbers More formally: –Given a predicate P and a domain D –The truth set of P is the set of elements from D that makes P to be true Copyright © 2014 Curt Hill

18. Exercises From 2.1 –3, 9,19, 23,27,43 Copyright © 2014 Curt Hill