1. 1 Section 1.6 Sets

2. 2 Set Fundamental discrete structure on which all other discrete structures are built Can be loosely defined as a collection of elements or members Elements do not have to be related, but usually are

3. 3 Describing Sets One method for describing sets is to simply list the members If members are related, can use ellipses to indicate a recurring pattern in a list For example, let M = {JAN, FEB, MAR, …, DEC} Sets are not required to be ordered; for example, suppose W = {DEC, MAR, JAN, FEB, APR, SEP, OCT, JUL, MAY, NOV, JUN, AUG} Since W contains the same elements as M, and ordering doesn’t matter, the two sets are said to be EQUAL

4. 4 Well-known Sets N is the set of natural numbers (positive integers) Z is the set of integers (both positive and negative) R is the set of real numbers Certain sets are used so commonly they are given symbols to represent them - the following are examples:

5. 5 Set Builder Notation Notation often used when all members of a set can’t be listed Characterizes all elements in a set by stating what property(s) make them members For example, we could use the following to describe the set of all students in this class: D = { x | x is a student in Discrete Math}

6. 6 Membership Notation We can use the symbol  to denote that some object is an element of a set; for example: 17  Z The symbol  indicates non-membership; for example: 1.3  Z

7. 7 The Empty Set The empty or null set is the set containing no elements The set may be denoted as { } or 

8. 8 Subsets A set A is a subset of set B if and only if every element in A is an element of B The symbol  is used to indicate a subset; for example, if E is the set of all even integers greater than 0, then E  Z The empty set is a subset of every set, and every set is a subset of itself

9. 9 Proper Subset Any set that is a subset of another, but is not equal to that set Can prove that 2 sets are equal by showing that each set is a subset of the other

10. 10 Cardinality If the number of elements in a set can be counted, then the set is finite A set that is not finite is infinite The number of elements in a finite set is the cardinality of the set For any set S, the cardinality of S is denoted as: |S|

11. 11 The Power Set The power set of a set is the set of all subsets of that set For set S, the power set is denoted P(S) For example, if S = {a,b,c,d} P(S) = { , {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {b,c,d}, {a,c,d}, {a,b,c,d}}

12. 12 Ordered n-tuples Sets are collections of unordered elements; a set of ordered elements is called an ordered n-tuple An ordered 2-tuple is also called an ordered pair 2 ordered n-tuples are equal if and only if they contain the same elements in the same order

13. 13 Cartesian Products Let A and B be sets. The Cartesian product of the 2 sets is the set of all ordered pairs (a,b) such that a  A and b  B For example, Let A = {Bob, Judy} and B = {Key West, San Francisco, Manhattan} The Cartesian product AxB is: {(Bob, Key West), (Bob, San Francisco), (Bob, Manhattan), (Judy, Key West), (Judy, San Francisco), (Judy, Manhattan)} B x A = {(Key West, Bob), (San Francisco, Bob), (Manhattan, Bob), (Key West, Judy), (San Francisco, Judy), (Manhattan, Judy)}

14. 14 Cartesian Products For n sets, the Cartesian product A 1 x A 2 x … x A n is the set of ordered n-tuples (a 1, a 2, …, a n ) where a i belongs to A i for i = 1, 2, … n Example: let A = {a,b}, B={c,d,e} and C={f} What is A x B x C? A x B x C = {(a,c,f), (a,d,f), (a,e,f), (b,c,f), (b,d,f), (b,e,f)}

15. 15 Section 1.6 Sets - ends -