Question

Prove set of postive rational numbers are group or not under addition

Answer 1

Let Q+ denote the set of all positive rational numbers.WE define an operation * on Q+as:

a*b=(ab)/2 V a,bϵQ+

To show that  (Q+,*) is a group.

CLosure Property:Since for every a,b ϵQ+,(ab)/2 is also in Q+therefore Q+ is closed with respet to operation *.

Assosiative PRoperty:Let a,b,c ϵ Q+.Then 

(a*b)*c=(ab/2)*c=[(ab)/2]c/2=a([(bc/2)]/2=a*(bc/2)=a*(b*c)

Commutativity:Let a,bϵQ+.Then a*b=(ab)/2=(ba)/2=b*a

Existense of identity:The number e will be identy element

if eϵQ+ and if e*a=a=a*e VaϵQ+

Now e*a=a

which implies (ea)/2 =a

a/2(e-2)=0

e=2 since aϵQ+ a is not equal to 0

now 2ϵQ+ and we have 2*a=(2a)/2=a=a*2 V a ϵQ+ 

Therefore 2 is the identity element.

Existense of inverse:Let a be any element of Q+If the number b is to be the inverse of a ,then we must have

b*a=e=2

(ba)/2=2

b=4/a

now aϵQ+

we have (4/a)*a {(4/a)a}/2=2=a*4/a      

therefore Thus each element of Q+is inversible.

Hence(Q+,*) is an ableain group.