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Union Intersection Likened to Logical Or and Logical And Relative Complement Absolute Complement Likened to logical Negation The union of two sets is the set that contains elements belonging to either of the two sets Equivalent to the Boolean operation “or” Written as: Examples: A = {a, b, c, d} B = {c, d, e, f} A B = {a, b, c, d, e, f} Note that the set could have been described as {a, b, c, d, c, d, e, f} A = {a, b, c, d} B = {c, d, e, f} Sets overlap A B = {a, b, c, d, e, f} A = {a, b, c, d} B = {x, y, z} Sets are A B = {a, b, c, d, disjoint x, y, z} The intersection of two sets is the set of all elements common to both sets The intersection of disjoint sets is the empty set Equivalent to the Boolean operation “and” written as: Examples: A = {a, b, c, d} B = {a, b} A B = {a, b} A = {a, b, c, d} B = {x, y, z} AB= A = {a, b, c, d} B = {a, b} A B = {a, b} A = {a, b, c, d} B = {x, y, z} AB= The relative complement (difference) of two sets is the set of elements contained in one, but not both, of the sets Related to the Boolean “Exclusive Or” Written as: — Examples: Given: A = {a, b, c, d} and B = {a, c, f, g} A — B = {b, d} B — A = {f, g} A = {a, b, c, d} B = {a, c, f, g } A — B = {b, d} B — A = {f, g} The absolute complement of a set is the set of elements which do not belong to the set being complemented’ Equivalent to the Boolean operation “not” Written as a superscripted ‘c’ Example: U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} and B = {a, b, c, d, e} Ac = {d, e, u, v, w} Bc = {u, v, w, x, y, z} U = {a, b, c, d, e, u, v, w, x, y, z} A = {a, b, c, x, y, z} B = {d, e, y, z} Ac = {d, e, u, v, w} Illustrates the 8 possible relations between Sets, A, B and C Region Relationship A B C 1 3 A B A B 4 A 5 A A A 2 6 7 8 C C B C B C B C B C A B C Shows whether an arbitrary element x belongs in any of the indicated sets. Sets A N N Y Y B N Y N Y Operations A B A B N N Y N Y N Y Y Idempotent Laws AA=A AA=A Associative Laws (A B) C = A (B C) (A B) C = A (B C) Commutative Laws AB=BA AB=BA Distributive Laws A (B C) = (A B) (A C) A (B C) = (A B) (A C) Identity Laws A=A AU=U AU=A A = Complement Laws A Ac = U A Ac = De Morgan’s Laws (A B)c = Ac Bc (A B)c = Ac Bc Uc = c=U