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Discrete Structures

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 . Subsets.

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Discrete Structures

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  1. 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 6.1 Set Theory - Definitions and the Element Method of Proof

  2. 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. 6.1 Set Theory - Definitions and the Element Method of Proof

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

  4. 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. 6.1 Set Theory - Definitions and the Element Method of Proof

  5. 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. 6.1 Set Theory - Definitions and the Element Method of Proof

  6. 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. 6.1 Set Theory - Definitions and the Element Method of Proof

  7. Set Equality • Given sets A and B, AequalsB, 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 6.1 Set Theory - Definitions and the Element Method of Proof

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

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

  10. 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) denotedB–A, is the set of all elements that are in Bbut not A. Symbolically: B – A= {x U | x  B and x  A} 6.1 Set Theory - Definitions and the Element Method of Proof

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

  12. 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 Ce. A C f. Bc g. Ac Bc h. Ac Bci. (A B)c j. (A B)c 6.1 Set Theory - Definitions and the Element Method of Proof

  13. 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, 6.1 Set Theory - Definitions and the Element Method of Proof

  14. 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 disjointiff they have no elements in common. Symbolically: A and B are disjoint  A  B =  6.1 Set Theory - Definitions and the Element Method of Proof

  15. Definitions • 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. 6.1 Set Theory - Definitions and the Element Method of Proof

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

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

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

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

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

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