G s x evenx interval notations a

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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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