1. Lecture 2.1: Sets and Set Operations CS 250, Discrete Structures, Fall 2014 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

2. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Course Admin HW1 Due 11am 09/18/14 – this Thursday Please follow all instructions Recall: late submissions will not be accepted

3. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Outline Set Definitions and Theory Set Operations

4. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation A set is an unordered collection of elements. Some examples: {1, 2, 3} is the set containing “1” and “2” and “3.” {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. {1, 2, 3, …} is a way we denote an infinite set (in this case, the natural numbers).  = {} is the empty set or null set, or the set containing no elements. U: is the set of all possible elements in the universe Note:   {  }

5. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation x  S means “x is an element of set S.” x  S means “x is not an element of set S.” A  B means “A is a subset of B.” Venn Diagram or, “B contains A.” or, “every element of A is also in B.” or,  x ((x  A)  (x  B)). A B

6. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation A  B means “A is a subset of B.” A  B means “A is a superset of B.” A = B if and only if A and B have exactly the same elements. iff, A  B and B  A iff, A  B and A  B iff,  x ((x  A)  (x  B)). So to show equality of sets A and B, show: A  B B  A

7. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation A  B means “A is a proper subset of B.” A  B, and A  B.  x ((x  A)  (x  B))   x ((x  B)  (x  A))  x ((x  A)  (x  B))   x  (  (x  B) v (x  A))  x ((x  A)  (x  B))   x ((x  B)   (x  A))  x ((x  A)  (x  B))   x ((x  B)  (x  A)) A B

8. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation Quick examples: {1,2,3}  {1,2,3,4,5} {1,2,3}  {1,2,3,4,5} Is   {1,2,3}? Yes!  x (x   )  (x  {1,2,3}) holds, because (x   ) is false. Is   {1,2,3}? No Is   { ,1,2,3}?Yes Is   { ,1,2,3}?Yes

9. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Definitions and notation Quiz time: Is {x}  {x}? Is {x}  {x,{x}}? Is {x}  {x,{x}}? Is {x}  {x}? Yes No

10. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Ways to define sets Explicitly: {John, Paul, George, Ringo} Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…} Set builder: { x : x is prime }, { x | x is odd }. In general { x : P(x) is true }, where P(x) is some description of the set. Ex. Let D(x,y) denote “x is divisible by y.” Give another name for { x :  y ((y > 1)  (y < x))   D(x,y) }. : and | are read “such that” or “where” Primes

11. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Cardinality If S is finite, then the cardinality of S, |S|, is the number of distinct elements in S. If S = {1,2,3}, |S| = 3. If S = {3,3,3,3,3}, If S = , If S = { , {  }, { ,{  }} }, |S| = 1.|S| = 0. |S| = 3. If S = {0,1,2,3,…}, |S| is infinite.

12. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Power sets If S is a set, then the power set of S is 2 S = { x : x  S }. If S = {a}, aka P(S) If S = {a,b}, If S = , If S = { ,{  }}, We say, “P(S) is the set of all subsets of S.” 2 S = { , {a}}.2 S = { , {a}, {b}, {a,b}}.2 S = {  }.2 S = { , {  }, {{  }}, { ,{  }}}. Fact: if S is finite, |2 S | = 2 |S|. (if |S| = n, |2 S | = 2 n )

13. 12/12/2015 Set Theory - Cartesian Product The Cartesian Product of two sets A and B is: A x B = { : a  A  b  B} If A = {Charlie, Lucy, Linus}, and B = {Brown, VanPelt}, then A,B finite  |AxB| = ? A 1 x A 2 x … x A n = { : a 1  A 1, a 2  A 2, …, a n  A n } A x B = {,,,,, } We’ll use these special sets soon! a)AxB b)|A|+|B| c)|A+B| d)|A||B| Lecture 2.1 -- Sets and Set Operations

14. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Operators The union of two sets A and B is: A  B = { x : x  A v x  B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Charlie, Lucy, Linus, Desi} A B

15. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Operators The intersection of two sets A and B is: A  B = { x : x  A  x  B} If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then A  B = {Lucy} A B

16. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Operators The intersection of two sets A and B is: A  B = { x : x  A  x  B} If A = {x : x is a US president}, and B = {x : x is in this room}, then A  B = {x : x is a US president in this room} =  A B Sets whose intersection is empty are called disjoint sets

17. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Operators The complement of a set A is: A = { x : x  A} If A = {x : x is bored}, then A = {x : x is not bored} A ==  = U and U =  U

18. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Set Theory - Operators The set difference, A - B, is: A U B A - B = { x : x  A  x  B } A - B = A  B

19. 12/12/2015 Lecture 2.1 -- Sets and Set Operations Today’s Reading Rosen 2.1