Answer to Question #143629 in Abstract Algebra for Dolly

Question

In the set R of real numbers define an algebraic operation α by
α(a,b)=a-b
(a) (R,α) is not group
(b) α is not commutative

Answer 1

a) A group is a set with an operation that has the following 4 properties:

1) The set is closed under the operation.

2) The set is associative under the operation.

3) The set has an identity element under the operation that is also an element of the set.

4) Every element of the set has an inverse under the operation that is also an element of the set.

 Real numbers are closed under subtraction.

We then have Subtraction on Real Numbers is not associative

$7-(4-3)=6\not=0=(7-4)-3$

Therefore (R,α) is not group .

b) $5-2=3\not=-3=2-5$

Therefore α is not commutative.