1. Relationships Between Sets

2. Union If we have two sets we might want to combine them into one big set. The Union of A and B is written We don’t bother to write the duplicate(s) twice. A = {1, 2, 3} and B = {3, 4, 5}

3. Union It’s just like real life! Two people get married (union) and merge their CD collections…

4. Union They sell all the duplicates on eBay. The CD’s that are left are the union of their collections. all of hers + all of his – dups

5. Union Mathematically, we say that the number of elements in the union of two finite sets is: the number of elements in Set A (his DVD’s) PLUS the number of elements in Set B (her DVD’s) MINUS the number of elements in the intersection of the sets (duplicates). The DVD’s that are left are the union of their collections. all of hers + all of his – dups

6. Intersection Just like the intersection of two roads, the intersection of two sets are the elements that are members of both sets.

7. Intersection Set A = { 1, 2, 3 } set B = { 3, 4, 5 }

8. The empty set Set A = { 1, 2, 3 } set B = { 4, 5, 6 } There is no element that is in both sets, so the intersection is the empty set.

9. The empty set The symbol for the empty set is the Greek letter Phi Do not put it in brackets. is not empty. It is a set with one element – the Greek letter Phi

10. Venn Diagrams One way to graphically represent sets is by using Venn diagrams. John Venn (1834 – 1923), was a British logician and philosopher who introduced the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science. Courtesy of Wikipedia

11. Universal Set To use a Venn diagram, you must first start with a universal set, represented by U, which contains all of the elements being considered in a problem. U = {all the puppies in a litter}

12. Universal Set In a problem about Chevys and Fords, the universal set could be the set of all cars or it could be the set of all trucks – it just has to include all of the vehicles that you’re going to be talking about.

13. Venn Diagram If U = {all cars} and F = {all Ford cars}, the Venn diagram would look like this: It’s easy to see that F is a proper subset of U. U F

14. Venn Diagram The complement of F is the set that consists of all of the members of U that are not in F The complement of F is written F’. The complement would be all cars that are NOT Fords. U F (the blue area)

15. Venn Diagram If U = {all cars}, and F = {all Ford cars}, and C = {all Chevy cars} the Venn diagram would look like this: It’s easy to see that both F and C are subsets of U. U F C

16. Disjoint Sets In this case, the sets are disjoint - meaning that they don’t overlap. U FC There are no cars that are both Fords and Chevys - my neighbor sort of has one, but that’s a long story…

17. Intersection R = {redheads} G = {people with green eyes} Venn diagrams make it easier to see how sets relate.

18. Vocabulary Union Intersection Empty set Venn Diagram