1. SETS A set B is a collection of objects such that for every object X in the universe the statement: “X is a member of B” Is a proposition.

2. A quick review of basic notation and set operations. 1. A = {1, 2, ab, ba, 3, moshe, table}, 2.1,2, ab, ba, moshe table are “elements.” They are members of the set A or “belong” to A. Notation: ab  A a  A 3. V = {a, i, o, u, e} Set of Vowels O = {1,3,5,7,9} Odd numbers < 10. 4. A 1 = {2, 5, 8, 11, …, 101} A 2 = {1, 2, 3, 5, 8, 13,…} A 3 = {2, 5, 10, 17, 26, …, 101}

3. 1. Set Builder: B = {x | P(x)} B 1 = {x | x = n 2 + 1, 1  n  10} (B 1 = A 3 ) B2 = {p | p prime, p = n! + 1, n  Special sets: N (non-negative integers, natural numbers) Q (rational numbers) Z (integers) Z + (positive integers) R (real numbers)  (the empty set) Basic notation.

4. Relations among sets 1. A = B: if x  A  x  B 2. Subsets: A  B A  B A  B A  B 3. For every set B:   B 4. A set may have other sets as members: A = { , {a}, {b}, {a,b}}. Note: A has 4 elements.   A and also   A, {{a}}  A, a  A, {a}  A.

5. Set operations Union: A  B = {x | x  A  x  B} (logic “or”) Intersection: A  B = {x | x  A  x  B} ( “and”) Set difference: A \ B = {x | x  A  x  B} Complement of A: A = {a | a  A} or if U is the “universe” then A = U \ A (“not”). Example: If U = {a,b,c,…,z} and A = {i,o,e,u,a} then A = {n | n is not a vowel}. Symmetric difference: A  B = (A \ B)  (B \ A) (“xor”)

6. The characteristic vector of a set (representing sets in memory): Let U = {1,2,…, 15}. Let A = { 3,5,11,13} the characteristic vector of A is the binary string 00101 00000 10100. The characteristic vector 10010 01101 10001 represents the set {1,4,7,8,10,11,15}. 00000 00000 00000 represents . Note: with this representation the union of two sets is the OR bit operation and the intersection is the AND.

7. A simple application. Problem: find the smallest integer n that satisfies the following 3 conditions simultaneously: (n mod 7 = 5), (n mod 11 = 7), (n mod 17 = 9) Knowing the language “Math” can help us look for information and use various systems to solve this problem. The following exolains how to use SAGE's set operations to solve problems.

8. We can create three sets: 1. A = {k | k = 7n + 5, k < 4000} 2. B = {k | k = 11n + 7, k < 4000} 3. C = {k | k = 17n + 9, k < 4000} We can then ask SAGE to find the intersection of the three sets. The smallest integer in the intersection (provided there is one) will be our solution. Answer: {502, 1811, 3120,...}

9. Venn Diagrams Venn Diagrams : a useful tool for representing information. For instance, the various sets that can be formed by the basic set operations can be viewed by a Venn Diagram.

10. A B C

11. Proving set equalities: Either: x  A  x  B or A  B  B  A. Example: De Morgan’s law: A  B = A  B Proof: Let x  A  B. Then: x  A  B. Or: x  A and x  B Or: x  A and x  B Or: x  A  B Conversely, start from the bottom and go up. QED Assume M = {1,2,5,9} then = A 1  A 2  A 5  A 9

12. Notation: A 1  A 2  …  A n = {x | x  A i i = 1, 2, …, n}. A 1  A 2  …  A n = {x |  ( i, 1  i  n) x  A i. Use formula to insert intersection. Assume M = {1,2,5,9} then = A 1  A 2  A 5  A 9

13. The Power Set Definition: The Power set of the set A is: P(A) = {B | B  A}.  has 0 elements. P(  ) has one element: P(  ) = {  } A = {a} P(A) = { , {a}} P({  }) = { , {  }}

14. The cartesian product Cartesian product : A x B = {(a,b) | a  A  b  B} Can be defined using sets only: A x B = {{a}, {a,b}| a  A  b  B} Note: (a,b)  (b,a) if a  b. Cartesian product of n sets: A 1 x A 2 x … x A n = {(a 1, a 2,…, a n ) | a i  A i, i = 1,…,n}

15. Relations Definition 1: A relation R, (binary relation) between two sets A and B is a subset of A x B (mathematically speaking: R  A x B). Definition 2: A relation R on a set A is a subset of A x A.

16. Relations There are two common ways to describe relations on a set or between two sets: List all pairs belonging to the relation. Use set builders to describe the pairs. Example 1: R 0 = {(4,3), (9,2), (3,6), (7,5)} is a relation on N. It is also a relation on A x B where A = {4,9,3,7} and B = {3,2,6,5}

17. More examples Example 2: R 2 = {(n,k) | n  N and n + k is a prime number}. Example 3: R 3 = {(n,k) | n,k  N and |n – k| is a multiple of 19}. Example 4: R 3 = {(w,m) | w is a woman, m is a man, w dates m}

18. Classification of relations These definitions apply to relations on A. Definition 3: A relation R on A is reflexive if (a,a)  R  a  A. Definition 4: A relation R on A is symmetric if (a,b)  R then (b,a)  R. R is antisymmetric if (a,b)  R and (b,a)  R only if a = b. Definition 5: A relation R on a set A is transitive if (a,b)  R  (b,c)  R then (a,c)  R.

19. The transitive closure Observation: If R 1 and R 2 are transitive relations on a set A then so is R 1  R 2. Proof: Obvious. Definition 6: The transitive closure of a relation R on a set A is the “smallest” transitive relation R* on A such that R*  R.

20. I think I solved it!