Abstract Algebra Answers

Q. Do the even integer form a group w.r.t addition.

Q. Do the non zero positive rational no. form a group w.r.t multiplication.

Q. Do the odd integer form a group w.r.t addition.

Assume that A and B are isomorphic commutative rings with unity. Prove that if A is a field, so is B.

16. 5 The following are true except (a) If every element of a set B is an element of a set A, then B is a subset of A. (b) A is a subset of B is denoted by A⊂B (c) Two sets A and B are equal if and only if A⊆B and B⊆A (d) Two sets A and B are equal if and only if A = -B and B = A
17. If A = { a + i b / a, b ϵ R }, then A is a set of ……… numbers (a) rational (b) natural (c) real (d) complex

Given a set X={a,b,c}, and a function Ψ:X→X define by Ψ(a)=b,Ψ(b)=a,Ψ(c)=c
. the function is (a)Only unto (b) only injective (c) bijective (d)no solution
A matrix X=(35 12 ) define a function from R 2 toR 2 by f X (a,b)=(3a+b,5a+2b) . find the inverse function of f X (a)f −1 X (a,b)=(3a−b,−5a+3b) (b)f 1 X (a,b)=(2a−b,−5a+3b) (c)f 5 X (a,b)=(2a−b,−5a+3b) (d) f 1 X (a,b)=(2a−b,−5a+4b)
Which of the following is divisible by 17 for all positive integer n (a) 7 n +2 (b) 6 n +2 (c) 2.7 n +3.5 n −5 (d)3.5 2n+1 +2 3n+1

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.