(Bonus 2.) We discussed early in the semester that arithmetic is well-defined on

Question

(Bonus 2.) We discussed early in the semester that arithmetic is well-defined on equivalence classes of rational numbers, e.g., that even if you replace by an equivalent form like In this problem you will show that the same is true for the equivalence classes of integers defined as in Problem 3. Suppose that (m,n), (r, s) E A. Addition is defined by and multiplication is defined by (m, n). (r, s):-(mr + ns, nr + ms). (a) Using the above definitions, calculate the sum and product of (5,3) and (10,7). Re- (b) Prove that addition is well-defined: suppose that (m',n')Rm, n) and (,)Rr,s) (e) Prove that multiplication well-defined: if (m, m)R(m,n) and (' )Rr-.s), show thet calling that (m,n) is meant to represent the integer "m -n", compare your calcula- tions to normal integer arithmetic. and show that 25

Explanation / Answer

(a)

(5,3) + (10,7) = (5+10, 3+7) = (15, 10)

(5,3) * (10,7) = (5*10+3*7, 3*10+5*7) = (50+21, 30+35) = (71, 65)

Given more questions in single Q&A post, as per guidelines we answered one only.

That no information give for relation R

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