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Set Operations MATH 102 Contemporary Math S. Rook Overview • Section 2.3 in the textbook: – Intersection & Union – Complement – Difference Intersection & Union Intersection of Sets • Given sets A and B, the intersection of A and B, denoted A B , means to list those elements common to both sets – Only those elements present in BOTH A and B are part of the intersection – e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find A B – Can also represent using a Venn Diagram Union of Sets • Given sets A and B, the union of A and B, denoted A B , means to combine the elements of A and B together – i.e. fill an initially empty set (bag) with the elements of A and then the elements of B – An element present in BOTH A and B is only added once to the union – e.g. Let A = {1, 2, 3, 4, 5} and B = {2, 4, 6, 8}. Find AUB – Can also represent using a Venn Diagram Union of Sets (Continued) • Given sets A and B, what is the relationship between n(A), n(B), and n(A U B)? – e.g. Let A = {p| p is a person sitting in the front row} and B = {g | g is a person wearing glasses}. What are the values of n(A) and n(B)? • What is the value of n(A U B)? • Why does the sum of n(A) and n(B) not equal n(A U B)? n A B n( A) n( B) n A B – Verify for yourself that the value of n(A U B) checks for sets A and B on the previous slide Intersection & Union (Example) Ex 1: Let A = {2, 3, 4, 5, 7, 9}, B = {x | x is an even natural number}, and C = {y | y is an odd natural number} . Find the following sets: a) A B b) B C c) B C Complement Complement • Universal set: the set of all elements being considered in a problem. Often denoted by U. – All subsets in a problem are taken from the universal set • Given set A, the complement of A, denoted by A’, means to list those elements that A is missing from the universal set – i.e. those elements that need to be added to A to complete the universal set – e.g. Let U = {1, 2, 3, … , 10} and A = {1, 6, 9, 10}. Find A’. – Can also represent using a Venn Diagram Complement (Example) Ex 2: Let U = {a, b, c, d, e, f, g, h}, A = {a, b, c, e, g}, B = {a, b, c, d, e, f, g, h}, and C = { }. Find: a) A’ b) B’ c) C’ Difference Difference • Given sets A and B, the difference of A and B, denoted A – B, means the resulting set when the elements of B are removed from the elements of A – i.e. Just like subtraction, we are taking away those elements in B away from A – e.g. Let A = {1, 2, 3, 4, 5, 6} and B = {2, 3, 4}. Find A – B. – Can also represent using a Venn Diagram Difference (Example) Ex 3: Let U = {1, 2, 3, … }, A = {1, 3, 5, 7, 9, … }, B = {1, 2, 3, 4, 5, 6, 8}, and C = {2, 4, 6, 8}. Find the resulting set: a) B – C b) C – B c) A – C Combining Set Operations (Example) Ex 4: Let U = {1, 2, 3, …, 10 }, A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5, 6}, and C = {2, 4, 6, 7, 8}. Find the resulting set: a) A 'B C ' b) A B C ' ' Combining Set Operations (Example) Ex 5: Shade the appropriate regions in a Venn Diagram to represent the resulting set: a) A B C b) A C B C ' Summary • After studying these slides, you should know how to do the following: – – – – – Find the intersection and union of sets Calculate the number of elements in the union of sets Find the complement and difference of sets Apply multiple set operations Use Venn Diagrams to illustrate set operations • Additional Practice: – See the list of suggested problems for 2.3 • Next Lesson: – Survey Problems (Section 2.4)