Define the following order on the set Z × Z: (a, b) < (c, d) if...

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Advance Math

Define the following order on the set Z × Z: (a, b) < (c, d)if either a < c or a = c and b < d. This is referred to asthe dictionary order on Z × Z.
(a) Show that there are infinitely many elements (x, y) in Z × Zsatisfying the inequalities (0, 0) < (x, y) < (1, 1).
(b) Show that Axioms O1–O3 ( Trichotomy, Transitivity, Additionfor inequalities) are satisfied for this ordering.
(c) Give an example that shows that Axiom O4 (Multiplication forinequalities) is not satisfied for this ordering.
(d) Is the well-ordering axiom satisfied for Z × Z with thedictionary order?

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3.9 Ratings (654 Votes)

Problem aConsider the setAs defined above is a infinite setAnd for any element wehaveThe first inequality is because and The second inequality is because So there are infinitely many elements such that Problem bBy definition if then either orIn any case we have Keeping this in mind let us proceedLaw oftrichotomy Let See Answer

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