Understanding Sets and Venn Diagrams: Union, Intersection, and Complement
This presentation covers essential concepts of sets using Venn diagrams. Key topics include the definitions and examples of set complement, intersection, union, and difference of two sets. Through practical examples with natural numbers, the relationship between these sets is illustrated, alongside the union rule for sets. Students will also find test problems to solidify their understanding of these foundational concepts in set theory. It is crucial to attend the class for diagram constructions, as they are not included here.
Understanding Sets and Venn Diagrams: Union, Intersection, and Complement
E N D
Presentation Transcript
Warning: All the Venn Diagram construction and pictures will be done during class and are not included in this presentation. If you missed class you should get class notes from another student. Sets Day 2
Venn Diagrams Example. Let U be the set of natural numbers less than or equal to 10. Let A={2,4,6} and B={1,2,3,4,5}. (Note: I constructed this Venn Diagram during class and referred to it throughout this presentation.)
Definition of Set Complement The complement of set A, symbolized by A′, is the set of all the elements in the universal set that are not in the set A. A′ is read “A complement,” or “A prime.”
Set Complement Example. Let U be the set of natural numbers less than or equal to 10. Let A={2,4,6} and B={1,2,3,4,5}. Find A′ and B′. Answers: A′ = {1,3,5,7,8,9,10} B′ = {6,7,8,9,10}
Definition of Set Intersection The intersection of sets A and B, symbolized by A∩B, is the set containing all the elements that are common to both A and B. A∩B is read “A intersect B,” or “A AND B.”
Set Intersection Example. Let U be the set of natural numbers less than or equal to 10. Let A={2,4,6} and B={1,2,3,4,5}. Find A∩B. Answer: A∩B = {2,4}
Example Let U = {a, b, c, d} A = {a, c} B = {b, d} 1. Find A′. 2. Find U′. 3. Find φ′. 4. Find A∩B. 5. Find (A∩B)′. Answers: A′ = B • U′ = φ • φ′= U • A∩B = φ • (A∩B)′= U
Definition of Set Union The union of set A and set B, symbolized by A∪B, is the set containing all the elements that are members of set A or of set B (or of both sets). A∪B is read “A union B,” or “A OR B.”
Set Union Example. Let U be the set of natural numbers less than or equal to 10. Let A={2,4,6} and B={1,2,3,4,5}. Find A∪B. Answer: A∪B = {1,2,3,4,5,6}
Example Let U = {a, b, c, d} A = {a, c} B = {b, d} 1. Find A∪B. 2. Find (A∪B )′. 3. Find A′∩B. 4. Find (A′∩B)′. Answers: A∪B = U (A∪B )′ = φ A′∩B = B (A′∩B)′ = A
Union Rule for Sets The relationship between sets A, B, A∪B, and A∩B is given by the union rule: n(A∪B) = n(A) + n(B) – n(A∩B) Why? (We used aVenn Diagram to show this.)
Two Possible Union Rule Test Problems: • If n(A) = 5, n(B) = 8, and n(A∩B) = 2, find n(A∪B). Answer: n(A∪B)=5+8-2=11 • If n(A)=12, n(A∪B)=22, and n(A∩B)=10, find n(B). Answer: 20
Difference of Two Sets The difference of two sets A and B, symbolized A-B, is the set of elements that belong to set A but not to set B. Example. Let U ={1,2,3,…,10}, A={2,4,6} and B={1,2,3,4,5}. Find A-B and B-A. A-B={6} B-A={1,3,5}
