Prove whether the given satifies the four group axioms. If not determine whether it is a semigroup or monoid. 1. <R, *> where a*b = `a^(2)+ b` 2. <z, *> where a*b = a + b + ab ` `
The four group axioms are closure, associativity of the binary operation, an identity element, and inverses.
(a) The set is closed under this operation. (Squaring a real number and adding a real number to the result will always give a real number.)
(b) Associativity: Let `a,b,c in RR`
The operation is not associative.
The set with the given operation is not a group. Further, it is neither a semigroup or a monoid as both requires an associative binary operation.
(a) The set is closed under this operation -- an integer plus an integer plus the product of integers is an integer.
(b) Associativity -- let `a,b,c in ZZ`
So the operation is associative
(c) The identity is 0
for any `a in ZZ`
(d) Inverse: Suppose b is the inverse of a with `a,b in ZZ` .Then
`==>b=(-a)/(1+a)` which is not an integer for every `a in ZZ` so inverses do not exist.
This set with the given operation is not a group.
Since the binary operation is associative it is a semigroup. Since there is an identity element it is a monoid.