Relationships Between Sets

Union Two people get married (union) and merge their DVD collections…

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

N(AUB) = n(A) + n(B) – n(A B) 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). N(AUB) = n(A) + n(B) – n(A B) The DVD’s that are left are the union of their collections. all of hers + all of his – dups

Union Set A = { 1, 2, 3 } Set B = { 3, 4, 5 }

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

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

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.

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

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.

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. In a problem about Chevys and Fords, the universal set might be the set of all cars.

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

The complement of F is written F’. Venn Diagram The complement of F is the set that consists of all of the members of U that are not in F (the green area) The complement of F is written F’. U F

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

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

Vocabulary Element Subset Union Intersection Empty set