1. ## Groups

Hey, I am not entirely sure if I am doing this right. But anyways...
The directions say...

"Determine whether the binary operation * defined on the given set results in a group."
(A group is classified by 3 properties:
Associative Law a * (b * c) = (a * b) * c
Existence of an Identity: There exists an element, e, such that
a * e = e * a = a
Existence of Inverses: For each element, a, there exists an element, s,
such that a * s = s * a = e (the identity)

So, here is what I'm supposed to do:

(a) Let * be defined on the positive reals by a * b= √(ab)

(b) Let * be defined on all the Reals except 0, by a * b= a/b

(c) Let * be defined on all the Reals except 0, by a * b= a+b+ab

Any help would be much appreciated!

2. Originally Posted by dude15129
Hey, I am not entirely sure if I am doing this right. But anyways...
The directions say...

"Determine whether the binary operation * defined on the given set results in a group."
(A group is classified by 3 properties:
Associative Law a * (b * c) = (a * b) * c
Existence of an Identity: There exists an element, e, such that
a * e = e * a = a
Existence of Inverses: For each element, a, there exists an element, s,
such that a * s = s * a = e (the identity)

So, here is what I'm supposed to do:

(a) Let * be defined on the positive reals by a * b= √(ab)

(b) Let * be defined on all the Reals except 0, by a * b= a/b

(c) Let * be defined on all the Reals except 0, by a * b= a+b+ab

Any help would be much appreciated!
Hi,

(a) Yes.

(b) No. Associativity fails.

(c) No. The identity element does not exist.

See if you can come up with the detailed work.

3. Awesome...Those were actually what I was thinking, but I wasn't really sure. Thanks so much!