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2 1. ,y 1) """(x2 ,y2 ) if and only if Xi+ y2 = X2 +Yi is an equivalence relation on N x N. 2. Confirm that Z does have the arithmetic and order properties (1 )-(7) listed above: 3. Prove thatfor any xeZ we must havex · 0 = 0, and that if x andy are any non-zero members of Z then x • y -:I= 0. 4. Show that Z has the upper bound property, namely, if A is any non-empty subset of Z which is bounded above in Z then A has a least upper bound (actually a greatest member) in Z. State and prove the dual lower bound property for Z.

Therefore we must always have 1 > 0. For brevity let us write n(l) to denote the sum 1+1+1+ ... es) . Since 1 is positive it follows that n(l) cannot be zero (since F + is closed under addition and does not contain 0). Therefore F must contain infinitely many distinct positive elements, 1, 2(1), 3(1), ... , n(l), ... together with their corresponding additive inverses -1, -2(1), -3(1), ... , -n(l), ... If we identify each element n(l) of the field Fwith the corresponding integer n, and similarly each inverse - n(l) with the negative integer - n, then it is plain that we can embed Z in F; more precisely, we can say that every non-trivial totally ordered .

If either (i) p + q < r + s, or (ii) p + q = r + s , and p < r. This gives the elements of X arranged in the form which is the same as we would get by traversing the doubly infinite array of all the elements of all the sets X,,, along successive diagonals as shown below: x, x, x, x, Thus X C N and so Xis countably infinite. • Corollaries. (i) The union of any finite collection of countably infinite sets is countably infinite. ) sets is countably infinite. • Infinite and countably infinite sets Sec.

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