1. Which of the following are monoids under arithmetic addition?

1. Which of the following are monoids under arithmetic addition? (i). Z, (ii). 2Z,(iii). 2Z + 1, (iv). 2N, (v). Z∪ 1/2 Z2. Let (M, *, e*) be a monoid. Let m ∈ M. Define a new product Ω*m in M byΩ*m : (a, b) → a*m b = a * m * b.Show that this defines a semigroup. Under what condition on m do we have a unit elemente*m relative to Ω*m?3. For any invertible element a in a monoid M, and for any n ∈ Z, show that(a^n)^-14. Show that an element a ∈ M is invertible if and only if there exists x ∈ M such thata * x * a = e*.5. Let Q denote the set of all rational numbers. Define an operation Ω* bya * b = a + b + (a · b),where + and · denote arithmetic addition and multiplication in Q, respectively. Showthat the system (Q,*) forms a monoid.6. Let Q+ denote the set of all positive rational numbers. Show that the system (Q+,*)forms a commutative semigroup if the operation ? is defined bya * b = a · b/ a + b,where + and · are as in the previous problem.7. Consider the set S = {a + b.√3: a,b∈Z}, where + and · are as in the previous problem.Show the S forms a monoid with respect to ·.8. Let R+ be a set of all positive real numbers. If we define, for all p1, p2 ∈ R+,Ω* : R+ × R+ → R+byΩ* : (p1, p2) → p1 * p2 = p1 + p2/1 + (p1 · p2) ,where + and · denote arithmetic addition and multiplication in R+, respectively. Showthat R+ is a semigroup with respect to Ω*. Is R+ also a monoid? If not, why?9. Let M = Z × Z be the set of pairs of integers (a, b). Define(a, b) * (a', b') = (aa' + 2bb', ab' + ba').Show that this defines a monoid.