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 ` `

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The four group axioms are closure, associativity of the binary operation, an identity element, and inverses.

(1) `{RR,*},a*b=a^2+b`

(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.

(2) `{ZZ,*},a*b=a+b+ab`

(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.