15 Slides--Sets

15 Slides--Sets - CS103 HO#15 Slides-Sets April 14, 2010...

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CS103 HO#15 Slides--Sets April 14, 2010 1 Sets Sipser: "A set is a group of objects represented as a unit. Sets may contain any type of object, including numbers, symbols, and even other sets." If A is a set and x is an object, we write x A to mean that x is an element of (or member of) A, and x A to mean that x is not an element of (or member of) A. Sets Another definition: "A set is an unordered collection of distinct elements." One way to specify a set is to list the elements: A = {2, 3, 5, 7, 11} An object is either in a set or not, so there is no point in repeating elements in the list, nor does the order matter: {2, 3, 5, 7, 11} is the same set as {11, 7, 5, 2, 3, 7, 11} Another way to specify a set is by providing a rule or property that determines membership: A = {n | Prime(n) n < 12} This is called set builder notation. Sets Sets A and B are equal if and only if they have exactly the same members: A = B ↔ ∀ x ((x A) (x B)) A is a subset of B means that every member of A is a member of B: (A B) ↔ ∀ x ((x A) (x B)) A is a proper subset of B means that A subset of B and A is not equal to B: (A B) ((A B) (A B) Sipser writes this as A B Sets Sets A and B are equal if and only if they have exactly the same members: A = B ↔ ∀ x ((x A) (x B)) ↔ ∀ x ((x A) (x B)) ∧∀ x ((x B) (x A)) (A B) (B A) Alternate definition: A = B ((A B) (B A)) The Empty Set We say that a set A is empty if x (x A) and we denote the empty set by { } or Ø Alternate: Ø = {x | x x} Is the empty set unique? Suppose that x (x Ø) and x (x Ö) Then x( ((x Ø) (x Ö)) ((x Ö) (x Ø)) ) since both conditionals are vacuously true. So Ø = Ö Union of two sets We say that union of two sets A and B is the set consisting of all elements that belong to A or B, and we denote the union operator by the symbol A B = {x | x A x B} Intersection of two sets We say that intersection
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15 Slides--Sets - CS103 HO#15 Slides-Sets April 14, 2010...

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