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Accelerated III (Acc. Precalculus) August 20, 2015 Sets What is a set? • A set is a well defined collection of objects. • Is S = {some numbers} a set? • Is W = {Fred, George, Ron} a set? What is a set? • A set is a well defined collection of objects. • No, S = {some numbers} is not well defined. • W = {Fred, George, Ron} is a set. What is in a set? • The objects in a set are called elements or members. • means “is a member of”. • We can write 6 {1, 2, 3, …} • Also, Harry W How do you write a set? • Sets can be written by rule S = {x| 0 < x < 10 and x Z} • or by roster S = {1, 2, 3, 4, …,9} How do you write a set? • They are enclosed by brackets and the elements are separated by commas. The series of dots (… ) are called ellipses and mean “the pattern continues”. More details... • Two sets are equal iff they contain the same elements. • If A = {2, 3, 4, … 11} and if B = {x|1 < x < 12 and x Z}, is A = B? • Yes, A = B. More details... • U usually stands for the universal set. • We are usually given the universal set. It is often a set of numbers, such as Z, W or R. More details... • The complement of a set (written A’ or Ā ) contains everything in the universal set that isn’t in the set itself. • If U = {a, b, c, … f} and if A = {f, a, c, e}, find A’. More details... • If U = {a, b, c, … f} is the universal set and • if A = {f, a, c, e}, • The complement of set A • which is written A’ = {b, d}. More details... • Set A is a subset of B ( A B) if and only if every member of A is also in B. • If B = {a, b, c, … f} and if A = {f, a, c, e}, then A B. • Is A’ B ? • Yes, A’ = {b, d}, so A’ B . More details... • Set A is a proper subset of B ( A B) if and only if A B but A B. • A = {a, c, e}, B = {f, a, c, e} and C = {a, c, e, f}, • Then A B, and A B. • Furthermore, B C, but B C. More details... • A set with no elements is called the empty set or null set and is written as {} or . • The empty set is a subset of all sets. More details... • The Power set of a set is the set of subsets of the set. • If A = {3, 5}, then P(A) = • { {3}, {5}, {3, 5}, } More details... • The cardinality of a set is the number of elements in a set, and is written |S|. • If K = {3, 5, 8}, then • |K| = 3 • What is |P(K)|? • |P(K)| = 8 More details... • The cardinality of a set can only be found if the set is finite. But infinite sets contain an infinite number of elements. More details... • The union of A and B (written AB) is the set that contains all elements of A and all elements of B. More details... • The intersection of A and B (written AB) is the set that contains all elements that A and B have in common. More details... • If the intersection of A and B is the empty set (written AB = ), then A and B are disjoint sets. More details... • The difference set of A and B (written A - B) is the set that contains all elements in A that are not in B. More details... • The set of elements that are either in A or B, but not both is called the symmetric difference of A and B and is written A B. More details... • The Cartesian Product of A and B, written A X B, is the set of all ordered pairs (a,b) where a A and b B. • So {1, 2} X {6, 8} = • {(1, 6), (1, 8), (2, 6), (2, 8)} Work These! • If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} • 1. Is N M ? • 2. Is L M? • 3. Is N M? Work These! • If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} • 4. Write N’. • 5. Write N O. • 6. Write N O. Work These! • If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} • 7. Write P(L). • 8. Write |O’|. • 9. Write M - O. • 10. Write M O. Work These! • If L = {1, 3}, M = {2, 3, 4}, N = {3, 4}, O = {2, 4, 5}, and U = {1, 2, 3, 4, 5} • 11. Write L X N. Any Questions?