Let ab with a0 b0 Prove that ab1a1b1Solutionto prove ab1 a1

Let a,b with a0, b0. Prove that (ab)^-1=a^-1b^-1.

Solution

to prove (ab)^-1 = a^-1b^-1

we know that for any c in R, if (ab)c = e , then c = (ab)^-1 & c is called inverse of (ab).

therefore we need to show (ab)(a^-1b^-1) = e

Consider (ab)(a^-1b^-1) = (ab)(b^-1a^-1) ...[commutativity holds in R]

= a(bb^-1)a^-1   ....[associativity]

= a.e.a^-1 ...[ bb^-1 = e ]

= aa^-1   = e

hence, (ab)^-1  = a^-1b^-1