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Unformatted text preview: = set of positive real numbers C = set of complex numbers Q = set of rational numbers r, where r = a/b and b ≠ 0 Set Builder Nota$on ༉ The property or properties that all members must satisfy: S = {x  x is a positive integer less than 100} O = {x  x is an odd positive integer less than 10} O = {x ∈ Z⁺  x is odd and x < 10} ༉ A predicate may be used: S = {x  P(x)}, e.g., S = {x  Even(x)} ༉ Interval Notations: [a, b = {x  a ≤ x ≤ b} (a, b ) = {x  a < x < b} [a, b ) = {x  a ≤ x < b} (a, b = {x  a < x ≤ b} Universal Set and Empty Set ༉ The universal set U is the set containing everything currently under consideration. ༉ Sometimes implicit. Venn Diagram ༉ Sometimes explicitly stated. ༉ Contents depend on the context. ༉ The empty set is the set with no elements. U V a b c d e ༉ Symbolized ∅ and {} used. ༉ The empty set is diﬀerent from a set containing ∅. ∅ = {} ≠ {∅} Set Equality Deﬁnition: two sets are equal if and only if they have the same elements. ༉ Therefore, if A and B are two sets, then A and B are equal if and only if they have the same elements: ༉ We write A = B if A and B are equal sets. {1, 3, 5} = {3, 5, 1} {1, 5, 5, 5, 3, 3, 1} = {1, 3, 5} Subsets Deﬁnition: the set A is a subset of B, if and only if every element of A is also an element of B. ༉ The notation A ⊆ B is used to indicate that A is a subset of the set B, and A ⊆ B holds if and only...
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 Spring '14
 M.Nojoumian
 Computer Science

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