Sec 2.3 Set Operations & Cartesian Products O Intersection of sets: A ∩ B is the set of elements common to both: A ∩ B = f x j x ∈ A and x ∈ B g A B U O Find the intersections of the following sets: f a, b, c g and f b, f, g g f a, b, c g and f a, b, c g f a, b, c g and f a, b, z g f a, b, c g and f x, y, z g f a, b, c g and ∅ © 2005-2009, N. Van Cleave 1 Disjoint Sets O Disjoint sets: two sets which have no elements in common. I.e., their intersection is empty: A ∩ B = ∅ A B U O Are the following sets disjoint? f a, b, c g and f d, e, f, g g f a, b, c g and f a, b, c g f a, b, c g and f a, b, z g f a, b, c g and f x, y, z g f a, b, c g and ∅ © 2005-2009, N. Van Cleave 2 Set Union O Union of sets: A ∪ B is the set of elements belonging to either of the sets: A ∪ B = f x j x ∈ A or x ∈ B g A B U Note: an element in the union of sets A and B may be a member of A, a member of B, or a member of both sets. O Find the unions of the following sets: f a, b, c g and f b, f, g g f a, b, c g and f a, b, c g f a, b, c g and f a, b, z g f a, b, c g and f x, y, z g f a, b, c g and ∅ © 2005-2009, N. Van Cleave 3 Set Difference O Difference of two sets: A – B is the set of all elements belonging to set A and not to set B. A – B = f x j x ∈ A and x∈ / B g A B U f 1, 2, 3, 4, 5 g – f 2, 4, 6 g = f 1, 3, 5 g but f 2, 4, 6 g – f 1, 2, 3, 4, 5 g = f 6 g Note: x∈ / B → x ∈ B0 (the complement of B) Thus, A – B = f x j x ∈ A and x∈ / B g = f x j x ∈ A and x ∈ B0 g = A ∩ B0 © 2005-2009, N. Van Cleave 4 O Given the sets: U = f1, 2, 3, 4, 5, 6, 9g A = f1, 2, 3, 4g B = f2, 4, 6g C = f1, 3, 6, 9g Find each of these sets: o A ∪ B = o A ∩ B = o A ∩ U = o A ∪ U = © 2005-2009, N. Van Cleave 5 U = f1, 2, 3, 4, 5, 6, 9g A = f1, 2, 3, 4g B = f2, 4, 6g C = f1, 3, 6, 9g o A0 = o A0 ∩ B = o A0 ∪ B = o A ∪ B ∪ C = o A ∩ B ∩ C = © 2005-2009, N. Van Cleave 6 O Describe each of the following sets in words: o A0 ∪ B0 = o A0 ∩ B0 = o A ∩ (B ∪ C) o (A0 ∪ C) ∩ B © 2005-2009, N. Van Cleave 7 O Given the sets: U = f1, 2, 3, 4, 5, 6, 7g A = f1, 2, 3, 4, 5, 6g B = f2, 3, 6g C = f3, 5, 7g Find each set: o A B = o B A = o (A B) ∪ C0 = Note, in general, A B =6 B A © 2005-2009, N. Van Cleave 8 Ordered Pairs O Ordered Pair: a group of two objects designated as first and second components. In the ordered pair (a, b): a is called the first component b is called the second component O In general (a, b) =6 (b, a), so order is important! O Two ordered pairs (a, b) and (c, d) are equal provided a = c and b = d (1, 3) = (1, 3) (1, 3) =6 (3, 1) (4, 9) = (4, 9) (9, 4) =6 (4, 9) (2+2, 33) = (22, 6+3) O Sets can contain ordered pairs: f (3, 3), (12, 6), (13, 29), (8, 7) g f (1, 3), (2, 6), (3, 9), . g © 2005-2009, N. Van Cleave 9 Cartesian Products O The Cartesian product of sets A and B is: A B = f(a,b) j a ∈ A and b ∈ Bg The Cartesian product of fa, b, cg f1, 2g = f (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2) g The Cartesian product of f1, 2g fa, b, cg = f (1, a), (1, b), (1, c), (2, a), (2, b), (2, c) g What's the difference between the two resulting sets above? If set A = fx, y, zg, what is A A ? © 2005-2009, N. Van Cleave 10 Cardinality of Cartesian Products O If set A has cardinality 5 and set B has cardinality 4, what is the cardinality of A B? Of B A? If jAj = n and jBj = m, what is jA Bj? © 2005-2009, N. Van Cleave 11 Set Operations O Finding intersections, unions, differences, Cartesian products, and complements of sets are examples of set operations O An operation is a rule or procedure by which one or more objects are used to obtain another object (usually a set or number). O Common Set Operations Let A and B be any sets, with U the universal set. o Complement of A is: A0 = fx j x ∈ U and x∈ / Ag A' A U © 2005-2009, N. Van Cleave 12 Set Intersection and Union o Intersection of A and B is: A ∩ B = fx j x ∈ A and x ∈ Bg A A B U B U o Union of A and B is: A ∪ B = fx j x ∈ A or x ∈ Bg © 2005-2009, N. Van Cleave 13 Set Difference o Difference of A and B is: A B = fx j x ∈ A and x∈ / Bg A B U © 2005-2009, N. Van Cleave 14 Cartesian Product o The Cartesian product of A and B is: A B = f(x,y) j x ∈ A and y ∈ Bg o Let U = fq, r, s, t, u, v, w, x, y, zg A = fr, s, t, i, vg B = ft, v, xg " Complete the Venn Diagram to represent U, A, and B A B U © 2005-2009, N. Van Cleave 15 o Shade the Diagram for: A ∩ B o Shade the Diagram for: (A0 ∩ B0) ∩ C A B A B C U U © 2005-2009, N. Van Cleave 16 o Shade the Diagram for: (A ∩ B)0 o Shade the Diagram for: A0 ∪ B0 A B Did we get these last two correct? U © 2005-2009, N. Van Cleave 17 De Morgan's Laws O De Morgan's Laws. For any sets A and B o (A ∩ B)0 = A0 ∪ B0 o (A ∪ B)0 = A0 ∩ B0 O Using A, B, C, ∩, ∪, , and 0, give a symbolic description of the shaded area in each of the following diagrams. Is there more than one way to describe each? A B A B U C U C © 2005-2009, N. Van Cleave 18.