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lect5 - Sets Different discrete structures are used in...

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1 Different discrete structures are used in modeling and problem solving: sets (unordered collections) relations (sets of pairs) graphs (sets of vertices and edges) Sets are used to group objects together. A set is a collection of objects (members, elements). A set is completely defined by its elements. An element ‘belongs to’ a set (or a set contains its elements). Notation: x A , x belongs to set A y A , y does not belong to set A Sets
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2 Ways to describe sets: 1. To list its elements in curly braces: a set of vowels in English alphabet V = {a, e, i, o, u} a set of odd positive integers less then 10 B ={1, 3, 5, 7, 9} a set of natural numbers (nonnegative integers) Ν = {0, 1, 2, 3, …} a set of integers: Z ={…,-2, -1, 0, 1, 2, …}
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3 2. To use set builder notation set of odd positive integers less then 10: B ={ x | x is positive odd integer less then 10} set of rational numbers: Q = { a / b | a , b Z and b 0 } set of positive integers: Z + = { x | x Z x> 0 } a set of squares of natural numbers: S ={ x 2 | x N } A = { ( a , b )| a , b N and a | b } 3. Special notation: an empty (null) set ={} or 2200 x [ x ]
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4 Definition. The set A is said to be a subset of B if and only if every element of A is also an element of B. A B 2200 x [ x A x B ] The subset A is said to be a proper subset of B if and only if B contains at least one element that is not in A. A B 2200 x [ x A x B ] ∧5 x [ x B x A ]
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5 Venn diagrams help visualize relation between sets B A x A B: 2200 x [ x A x B ] y A B: 2200 x [ x A x B ] and 5 y [ y B y A ] U - universe (of discourse), a set that contains all elements under consideration. U
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6 Theorem . Empty set is a subset of every set, that is ∅⊆ S Proof . We need to prove that any x that belongs to belongs to S, i. e. 2200 x [ x ∈∅ x S ]. But the hypothesis x ∈∅ is always false by the definition of empty set. So, the implication x ∈∅ x S is vacuously true.
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7 Do not confuse and (or ) signs! a A { a } A a A { a } A { a } A { b } A { b } A { a , { b }} A { a , { b }} A A A { } A Let A ={ a , , { b }}. T F F T T F T T T T T T
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8 Equality of two sets Definition . Two sets are equal if and only if each of them is a subset of the other. A = B A B B A Two sets are considered to be equal if they contain the same elements. For example: {1, 3, 5} = {3, 5, 1, 5} {1, 4, 9, 16} = { x 2 | x Z + x 2 <20} More rigorously:
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9 Set operations The union of two sets contain elements which belong to either of them or to both: } | { B x A x x B A
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