Question #1ba91

Question

1844 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-13T11:42:46

The Inverse of $2$ , denoted by, $2^-1$ , under the

operation $ast$ is $12 in ZZ.$

We assume that the Binary Operation $ast$ is a function,

$ast : ZZxxZZ to ZZ : a ast b=a+b-7, AAa,b in ZZ.$

Now, to find the inverse of $2$ w.r.t. $ast$ , we have to first find the

Identity Element , say, $e$ for $ast$ .

By the Defn. of $e$ , then, $aaste=easta=a, AA a.$

$aaste=a rArr a+e-7=a rArr e=7.$

Also, $7 ast a=a rArr 7+a-7=a.$

So, $7$ is the Identity for $ast$ .

Now, suppose that, $x$ is an Inverse of $2" under "ast$ .

Then, by Defn. of an inverse, we must have,

$x ast 2=2 ast x=e=7.$

$x ast 2=7 rArr x+2-7=7 rArr x=12.$

We also have, $2 ast 12=2+12-7=7=e.$

Thus, $12 ast 2= 2 ast 12=e(=7).$

Therefore, the Inverse of $2$ , denoted by, $2^-1$ , under the

operation $ast$ is $12 in ZZ.$

N.B.: In fact, it is easy to see that, $AA a in ZZ, a^-1=14-a.$

Enjoy Maths.!