Define a cyclic group, Mathematics

Assignment Help:

Question 1:

(a) Show that, for all sets A, B and C,

(i) (A ∩ B) c = Ac∩Bc.

(ii) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

(iii) A - (B ∪ C) = (A - B) ∩ (A - C).

(b) Let f: X → Y be a function and Ai ⊂ X for i ∈ I. Show that

(i) f(∩Ai) _∪f(Ai).

(ii) f(∪Ai) = ∪f(Ai).

Question 2:

(a) Prove that the set G = {a + b√3 : a; b ∈ Z} forms an abelian group under ordinary addition.

(b) Let R be a relation in the set of all natural numbers N defined by " xRy if and only if x - y is divisible by 3." Show that R is an equivalence relation. Hence determine the equivalence classes of R.

Question 3:

(a)    A set G consists of ordered pairs (a, b) such that a, b are real numbers and a ≠ 0. An operation * is defined in G as follows

(a, b) * (c, d) = (ac, ad + b) where a, b, c, d are real numbers. Prove that the given set constitutes a non-abelian group for the given operation.

(b) Let (G,*) be a group. Prove the following:

(i) the identity element e is unique,

(ii) (a * b)-1 = b-1 * a-1 for a, b ∈ G.

Question 4:

(a) Prove that a non-empty subset H is a subgroup of (G,*) if and only if

(1) a, b ∈ H ) a* b ∈ H.

(2) a ∈ H ) a-1 ∈ H where a-1 is the inverse of a in G.

(b) Prove that the set Q+ of all positive rational numbers forms an abelian group for the operation * defined by a * b = ab/ 2.

Question 5:

(a) (i) Define a cyclic group.

(ii) Prove that any cyclic group is abelian.

(iii) The set G = {1,-1, I,-i} forms a cyclic group under multiplication of complex numbers.

  • State a generator of G and show how it generates each element of G.
  • Draw the Cayley table for G.

(b) Let H and K be two subgroups of a group G. Show that H ∩ K is a subgroup of G.

 


Related Discussions:- Define a cyclic group

Scalar equation of plane - three dimensional spaces, Scalar Equation of Pla...

Scalar Equation of Plane A little more helpful form of the equations is as follows. Begin with the first form of the vector equation and write a vector for the difference. {

Find the probability density function, 1. The lifetime T (in days) of an el...

1. The lifetime T (in days) of an electrical component has reliability function given by: R(t) = e -0.01t for time t > 0. An electrical system consists of four such components. Th

Confidence interval, Confidence Interval The interval estimate or a 'co...

Confidence Interval The interval estimate or a 'confidence interval' consists of a range as an upper confidence limit and lower confidence limit whether we are confident that a

Addition and subtraction, In addition and subtraction we have discussed ...

In addition and subtraction we have discussed 1) Some ways of conveying the meaning of the operations of addition and subtraction to children. 2) The different models o

Algorithm for division, Also, their inability to apply the algorithm for di...

Also, their inability to apply the algorithm for division becomes quite evident. The reason for these difficulties may be many. We have listed some of them below. 1) There are n

Algebra, simplify mn+mp+nq+pq /n+p

simplify mn+mp+nq+pq /n+p

Determine if following sequences are monotonic or bounded, Determine if the...

Determine if the following sequences are monotonic and/or bounded. (a)   {-n 2 } ∞ n=0 (b) {( -1) n+1 } ∞ n=1 (c) {2/n 2 } ∞ n=5 Solution {-n 2 } ∞ n=0

What is the larger dimension in inches of the frame, Jessica has a picture ...

Jessica has a picture in a frame with a total area of 288 in2. The dimension of the picture without the frame is 12 in through 14 in. What is the larger dimension, in inches, of th

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd