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**Unformatted text preview: **The Real and Complex Number Systems Ordered Sets 1.5 Definition Let S be a set. An order on S is a relation, denoted by < , with the following two properties: (i) If x ∈ S and y ∈ S then one and only one of the statements x < y , x = y , y < x is true. (ii) If x,y,z ∈ S , if x < y and y < z , then x < z . 1.6 Definition An ordered set is a set S in which an order is defined. 1.7 Definition Suppose S is an ordered set, and E ⊆ S . If ( ∃ β ∈ S )( ∀ x ∈ E )( x ≤ β ), the E is bounded above , and β is an upper bound of E . Lower bounds are defined in the same way (with ≥ in place of ≤ ). 1.8 Definition Suppose S is an ordered set, E ⊆ S , and E is bounded above. Suppose there exists an α ∈ S with the following properties: (i) α is an upper bound of E . (ii) If γ < α then γ is not an upper bound of E . Then α is called the least upper bound of E or the supremum of E , and we write α = sup E . The greatest lower bound , or infimum , of a set E that is bounded below is defined in the same manner: The statement α = inf E means that α is a lower bound of E and that no β with β > α is a lower bound of E . 1.10 Definition An ordered set S is said to have the least-upper-bound property if the following is true: If E ⊆ S , E is not empty, and E is bounded above, then sup E exists in S . 1.11 Theorem Suppose S is an ordered set with the least-upper-bound property, B ⊆ S , B is not empty, and B is bounded below. Let L be the set of all lower bounds of B . Then α = sup L exists in S , and α = inf B . In particular, inf....

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