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Set Operations Chapter 2 Sec 3 Union What does the word mean to you? What does it mean in mathematics? Definition The union of sets A and B, written A B , is the set of elements that are members of either A or B(or both). Using set-builder notation, A B {x:x is a member of A or x is a member of B}. The union of more than two sets is the set of all elements belonging to at least one of the sets. Example Find the union of the following pair of sets. ◦ A={1,3,5,8,9} and B={2,4,6,7,8,} Solution A B The = {1,2,3,4,…,9} solution in the final answer we list the elements in order and did not list duplicate elements because doing so does not affect set equality. Intersection The essential idea of intersection is a region that is common to both sets. Definition The intersection of sets A and B, written A B , is the set of elements common to both A and B. Using set-builder notation, A B ={x:x is a member of A and x is a member of B}. The intersection of more than two sets is the set of elements that belong to each of the sets. If A B = , then we say that A and B are disjoint. To help us understand we will use the Venn Diagram r2 r1 r3 r4 A B r2 r3 r4 Inside the 2 circles are shaded. A B r2 r3 r4 Which region would be shaded? Complement If A is a subset of the universal set U, the complement of A is the set of elements of U that are not elements of A. This set is denoted by A’. Using set builder notation, A' {x : x U ; x A} Venn diagram: Complement r1 r2 What would be the complement? Example Find the complement of each set. ◦ U ={1,2,3,…10} and A ={2,4,6,8} A’ = {1,3,5,7,9} ◦ U is the set of cards in a standard 52 card deck, and F is the set of face cards. F’ is the set of nonface cards. Set difference The difference of sets B and A is the set of elements that are in B but not in A. This set is denoted by B-A. Using set builder notation, ◦ B-A = {x:x is a member of B and x is not a member of A}. Venn for set difference, B-A A r2 B r3 r4 Which region would be shaded? Examples of Set difference. Find the set difference. ◦ Find {3, 6, 9, 12}-{x:x is an even integer} ◦ Therefore we will remove all the even integers to get {3, 9}