Discrete Structures Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof The introduction of suitable abstractions is our only mental aid to organize and master complexity. – E. W. Dijkstra, 1930 – 2002 Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Subsets Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement: A  B  x, if x  A then x  B. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Subsets The negation is existential A  B  x, if x  A and x  B. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Subsets A proper subset of a set is a subset that is not equal to its containing set. A is a proper subset of B  A  B, and there is at least one element in B that is not in A. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Element Argument Let sets X and Y be given. To prove that X  Y, Suppose that x is a particular but arbitrarily chosen element of X, Show that x is an element of Y. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Example – pg. 350 # 4 Let A = {n | n = 5r for some integer r} and B = {m | m = 20s for some integer s}. Is A  B? Explain. Is B  A? Explain. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Set Equality Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically, A = B  A  B and B  A Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Operations on Sets Let A and B be subsets of a universal set U. 1. The union of A and B denoted A  B, is the set of all elements that are in at least one of A or B. Symbolically: A  B = {x  U | x  A or x  B} Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Operations on Sets Let A and B be subsets of a universal set U. 2. The intersection of A and B denoted A  B, is the set of all elements that are common to both A or B. Symbolically: A  B = {x  U | x  A and x  B} Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Operations on Sets Let A and B be subsets of a universal set U. 3. The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A. Symbolically: B – A = {x  U | x  B and x  A} Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Operations on Sets Let A and B be subsets of a universal set U. 4. The complement of A denoted Ac, is the set of all elements in U that are not A. Symbolically: Ac = {x  U | x  A} Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Example – pg. 350 # 11 Let the universal set be the set R of all real numbers and let A = {x  R | 0 < x  2}, B = {x  R | 1  x < 4}, and C = {x  R | 3  x < 9}. Find each of the following: a. A  B b. A  B c. Ac d. A  C e. A  C f. Bc g. Ac  Bc h. Ac  Bc i. (A  B)c j. (A  B)c Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

Unions and Intersections of an Indexed Collection of Sets Given sets A0, A1, A2, … that are subsets of a universal set U and given a nonnegative integer n, Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

Definitions Empty Set A set with no elements is called the empty set (or null set) and denoted by the symbol . Disjoint Two sets are called disjoint iff they have no elements in common. Symbolically: A and B are disjoint  A  B =  Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Mutually Disjoint Sets A1, A2, A3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets Ai and Aj with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, … Ai  Aj =  whenever i  j. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Example – pg. 305 # 23 Let for all positive integers i. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Partition A finite or infinite collection of nonempty sets {A1, A2, A3, …} is a partition of a set A iff, A is the union of all the Ai The sets A1, A2, A3, …are mutually disjoint. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Example – pg. 351 # 27 Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Power Set Given a set A, the power set of A is denoted (A), is the set of all subsets of A. Erickson 6.1 Set Theory - Definitions and the Element Method of Proof

6.1 Set Theory - Definitions and the Element Method of Proof Example – pg. 351 # 31 Suppose A = {1, 2} and B = {2, 3}. Find each of the following: Erickson 6.1 Set Theory - Definitions and the Element Method of Proof