For all x,y element of real number, define * by x*y=x+y-4?

Question

Show that * is a binary operation on the Real number Show that is a group Is abellian? Note: R is the real number

? · Accepted Answer

CLOSURE: * is a binary operation on the set of real numbers since x*y = x+y-4 is a real number for all real numbers x and y.

ASSOCIATIVITY:

Let x, y, z, be real numbers.

(x*y)*z = (x+y-4)*z = (x+y-4)+z-4 = x+y+z-8

x*(y*z) = x*(y+z-4) = x+(y+z-4)-4 = x+y+z-8

So (x*y)*z = z*(y*z) for all real numbers x, y, and z

IDENTITY ELEMENT

4 is the identity element.

Proof:

Let x be a real number. Then

x*4 = x+4-4 = x and

4*x = 4+x-4 = x

So x*4 = 4*x = x for all real numbers x.

INVERSE ELEMENT

The inverse element of x is 8-x.

Proof

Let x be a real number. Then

x*(8-x) = x + (8-x) - 4 = 4

(8-x)*x = (8-x) + x - 4 = 4

So x*(8-x) = (8-x)*x = 4 where 4 is the identity element.

So <R,*> is a group.

Let x and y be real numbers. Then

x*y = x + y -4 = y + x - 4 = y*x

So <R, *> is an abelian group.

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