Abstract Algebra Answers

If R (the set of real number) be the universal set and sets V={yϵR:0< y ≤3} and W={yϵR:2 ≤ y <4 } What is V l
For sets A and B , if A and B are subset of Z (the set of Integer) which of the following relations between the two subset is true? (a) (AuB)= A (b) (A\B)n(B\A)= empty set (c) (A\B)n(B\A)= Z (d) (A\B)u(B\A)= empty set
Which of the following pair of functions has f o g = g o f (a)f(y)=y 3 and g(y)=y √ 3 (b) f(y)=y^{5} and g(y)=3y+7$$ (c) f(y)=y 2 and g(y)=y+7 (d) f(y)=y 2 and g(y)=3y+7

let N is a normal cyclic subgroup of a group G,then prove that every subgroup of N is normal in G.

Show that subgroup and factor group of nilpotent group is nilpotent?

Let M and N be two subgroups of a soluble group G. Then show that MN is also soluble.

Mathematical Identities are sometimes indicated by the triple bar ----- when and why ??

Q. Let A and B be two ideals of ring R such that R=A⊕B show that R/A ≅B (R factor A is isomorphic to B) and
R/B ≅A (R factor B is isomorphic to A).

Give counter example to show that Every GCD (Greatest Common Divisor) Domain is not PID(Principal Ideal Domain)?