Let (A,\bullet) be a monoid sush that for every x in A, x\bullet x= where e is the identity element. Show that (A, ) is an abelian group.

8. Let 𝐺 be a group ith identity 𝑒 suoh that 𝑎2 = 𝑒 for all 𝑎 ∈ 𝐺, then 𝐺 is abelian.

let f:X -> (0,inf) be a continuous function on a compact space X. show that there exits an ε >0 such that f(x)>=ε for all x element of X

8. (a) Let G b e the group Z2 ⊕ Z2. Show that G is not y li . [5 marks℄(b) Let G b e the group Z3 ⊕ Z2. Show that G is y li . [4 marks℄

3. Let random variable X has exponential distribution with parameter ν.Find probabilityP (X ∈ [E(X) − σX ; E(X) + σX ])where σX is standard deviation of X.

Let g: E→F be a function defined by g(x) = 2x+1 where E={2,3,4}. What is g(E)? a. {5,7,9} b. {3,5,7} c. {4,6,8} d. {1,3,5}