Question

(6 pts) Alternate construction of the integers from the natural numbers. Suppose that the natural numbers N = {0,1,2, ...} ations. We do not yet have a notion of subtraction or the cancellation law for addition (if x+y = x+ z, then y = 2) and for multiplication given with the usual addition and multiplication oper negative numbers, though we do have are Define a relation R on N2 as follows (a, b) R (c, d) if and only if a + d = b+ c (a) Prove that R is an equivalence relation (b) Describe the equivalence classes of R, and how they correspond to the integers (c) Subtraction in the integers can be performed by adding the negative of the subtrahend1 i.e., 5 3 5(-3). How is a pair in N2 negated? (d) Define a multiplication on pairs in N2 that gives the usual multiplication of integers This alternate construction allows subtraction to be defined using only the usual addition in the natural numbers. Computers also can only add (they have no natural way to subtract) However, the representation of negative numbers used in computers (called two's complement) is different and relies on the fact that computers can only represent a finite range of numbers

Answer 1

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