I'm having some trouble understanding why cuts are defined with property 3 below:(adsbygoogle = window.adsbygoogle || []).push({});

"A cut in Q is a pair of subsets A, and B of Q such that

(a) [itex]A\cup B =Q[/itex], [itex]A\ne ∅[/itex], [itex]B \ne ∅ [/itex], [itex]A\cap B = ∅ [/itex]

(b) If [itex] a \in A [/itex] and [itex] b \in B [/itex], then [itex] a < b [/itex]

(c) A contains no largest element." (Pugh, 2001)

I'm not quite sure why property three must be in place. Why does A have "no largest element" or I guess, what does it mean by "no largest element" in this context?

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# Defining Cuts in the Rationals

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