1. Lecture 5: Set Theory 1 Dr Andrew Purkiss-Trew Cancer Research UK a.purkiss@mail.cryst.bbk.ac.uk Mathematics for Computing

2. Material to be covered today Set Theory 1 What are sets? How are they represented? Special and subsets Set Operations Power Sets Cartesian Products

3. What are Sets? A set is a well defined group of items. Sets are made up of elements e.g. The set of students studying at Birkbeck College

4. Set representation 1 Enumerated form: {2,4,6,8}Positive even numbers <10 {2,4,6,8,…,50} Positive even nos. <=50 {2,4,6,8,…}Positive even numbers {2,3,5,7,11,13,17,19} Prime numbers <20

5. Set Representation 2 Predicate form {x:x is even and 0 < x <= 50} {x:P(x)} Letters can represent sets A = {1,2,3,4,5} B = {x:x is a multiple of 2}

6. Set Representation 3 A = {1,2,3,4,5}, B = {x:x is a multiple of 2} Representation of elements 3  A, 2  B 6  A, 3  B Special Sets: N = {1,2,3,4,…} J = {…,-3,-2,-1,0,1,2,3,…} Q = {x: x = m/n for the integers m and n} R is the set of real numbers

7. Special Sets The null set:  enumerated form {} or predicate form {x: x  x} The universal set:  examples:  = R,  = J

8. Subsets Two sets A and B B is defined as a subset of A (represented B  A), when all elements of B are also elements in A. Example: A = {1,2,3,4,5,6}, B = {2,3,5}, C = {2,4,6,8}. B  A but as 8  C, but 8  A, C is not a subset of A.

9. Set Representation 4 A={1,2,3,4,5,6},B={2,3,5},C={2,4,6,8}  3 5 2 4 6 8 1 A B C 7 Venn Diagrams

10. More on subsets Another example: N  J  Q  R Other points For any set A, A  A and  A

11. Set equality Two sets A and B Definition A = B if A  B and B  Implications 1) {1,2,3} = {3,1,2} = {3,2,1} = {2,1,3} 2) {a,a,b} = {a,b}

12. Proper Subset B is a proper subset of A if: B  A and B  A.

13. Set operations Union A  B IntersectionA  B ComplementĀ DifferenceA – B

14. Union A  B = {x:x  A or x  B} A B

15. Intersection A  B = {x:x  A and x  B} A B

16. Complement Ā = {x:x  and x  } A

17. Difference A - B = {x:x  A and x  B} AB

18. Difference 2 A B A - B = A  ¯ B

19. Cardinality Cardinality. The number of elements in the set A = {1,2,3,4,5}, |A| = 5 B = {2,4,6,…,20}, |B| = 10

20. Power sets If A is a set, the power set of A,  (A) is the set of all subsets of A A = {1,2,3},  (A) = { , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

21. Cartesian Products A x B = {(x,y): x  A and y  B} Example: A = {1,3,5}, B = {2,4} A x B = {(1,2),(1,4),(3,2),(3,4),(5,2),(5,4)}

22. Home time End of Set Theory 1