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Use induction to prove that if b belongs to a ring and m is a positive...

Use induction to prove that if b belongs to a ring and m is a positive integer, then m(−b) = −(mb).

Notice that -(mb) is the additive inverse of mb, so mb+m(-b)=0. Also keep in mind that m is not a ring element

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Answer #1

At first we put m=1, then we assume m=p and finally we prove this is true also m=p+1.this is the step of induction.

Now put m=1, then 1.(-b)=-(1.b)=-b. Hence this is true for m=1.

Now we assume this is true for m=p, .then we have P (-b)= - (P. b),where P is a integer.

Now we prove, (P+1).(-b)= --((P+1).b).

L.H.S =(P+1).(-b)

=P. (-b) +(-b)

= - (Pb) +(-b)

= -( (P+1)(b)).=R.H.S.

Since the formula is true for m=1,p and also p+1, hence by induction we say that m. (-b) = - (mb).

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