Prove: R - {0}, the multiplicative inverse a^-1 is unique. Prove: a, b R, a(-b)

Question

Prove: R - {0}, the multiplicative inverse a^-1 is unique. Prove: a, b R, a(-b) = -(ab). You are only allowed to use the properties in the box on page 3 and properties 1 and 2 in the box on page 4. Make sure that your proof is syntactically complete, with a justification for each step. Prove: a, b R, |a - b| = |b - a| using the properties up to and including those in the box at the top of page 9. Make sure that your proof is syntactically complete, with a justification for each step. Do not just justify this based on the definition of distance.

Explanation / Answer

8) If possible let there be two multiplicative inverses for a as a-1 and b

Then a* a-1 = a*b = e = 1 identity element

This implies a-1 = 1/a and b = 1/a

Hence there can be only one inverse multiplicative

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a(-b) = a(0-b) (using additive identity)

= a*0-a*b(using distributive law)

= 0-ab (using 0 multiplication)

=-(ab)

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10) |a-b| = 0, if a=b

= a-b, if a>b

b-a , if b>a (by definition of modulus)

Consider |b-a| = 0, if a=b

= -(b-a) = a-b, if b<a

= b-a, if b>a

Comparing the two above we can conclude |a-b|=|b-a|

10)

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