NEB Class 12 Questions: NEB 12 Model Questions 2080/2024


NEB Class 12 Routine 2080-2081: Class 12 Routine

Elementary Group Theory - Exercise 3.3 : Class 12 Math

Complete Exercise 3.3 Elementary Group Theory. Exercise 3.2: Groups Complete Que Ans note, Class 12 Mathematics.
Elementary Group Theory - Exercise 3.3 : Class 12 Math

Chapter 1: Elementary Group Theory.


Exercise: 3.3

 

Complete Exercise of Elementary Group Theory - Exercise 3.3 : Class 12 Mathematics 2080 NEB.

Exercise 3.2 is about: Group and its types: Finite and infinite group, Trivial group, Abelian group Groups with elements other than numbers, Matrix Group are also include in this chapter.


Read: NEB Class 12 Mathematics All Chapter Exercise



Elementary Group Theory Exercise 3.3 Questions

Elementary Group Theory - Exercise 3.3 Questions : Class 12 Math

Elementary Group Theory - Exercise 3.3 Questions : Class 12 Math



Question no. 1

a.

Soln:

It is a false statement because the order of group G = Number elements in G = 4, which is finite.

 

b.

It is a true statement because the group contains two elements, so the order of given group is 2.

 

c.

Soln:

If is a false statement because it is not closed, for e.g.

(- 2).2 = - 4 ∉ {-2,-1,0,1,2}

 

d.

Soln:

It is false statement because it is not closed,

For e.g. 4 + 8 = 12 ∉ {2,4,6,8,10}

 


Question no. 2

Soln:

Here, 1 + 1 = 2 ∉ S where, 1 , 1 ԑ S.

So, S is not closed under addition.

Thus, S is not a group under addition.

 


Question no. 3

Soln:

Existence of Identity: Let a be any natural number and if it exists, let e be the identity element of a, then a + e = a. à e = a – a = 0 which is not a natural number,

So, identity element under addition doesn’t exist in the set of natural numbers,

Hence, the set of natural numbers does not form a group under the addition operation,

 


Question no. 4

Soln:

Closure property: From the multiplication table, we see that T is closed under multiplication, i.e.

Multiplication Table.

X

-1

1

-1

1

-1

1

-1

1

 

(-1) * (-1) = 1, (-1) * 1 = -1

1 * (-1) = - 1, 1 * 1 = 1

So, a * b ԑ T for all a,b ԑ T.

Associative property: Here, the elements of T are – 1 and 1. We know that all integers under multiplication obey associative law. So, the element of T being integers satisfy associative law under multiplication.

i.e. a * b(b * c) = (a * b) * c for all a,b,c, ԑ T.

For e.g. (-1) * {1 * (-1)} = (-1) * (-1) = 1.

And {(-1) * 1} * (-1) = (-1) * (-1) = 1

So, (-1) * {1 * (-1} = {(-1) * 1} * (-1) and so on.

Existence of Identity element,

Since (-1) * 1 = (-1) = 1 * (-1)

And 1 * 1 = 1

So, 1 is the multiplicative identity in T.

Existence of inverse

Each element of T is an inverse of itself since,

1 * 1 = 1 and (-1) * (-1) = 1.

Hence, T = {-1,1} forms a group under multiplication.



Question no. 5

a.

Soln:

Multiplication Table.

X

1

-1

i

-i

1

1

-1

i

-i

-1

-1

1

-i

i

i

i

-i

-1

1

-i

-i

i

1

-1

 

From above table, we see that for al a,b ԑ G, a * b ԑ G. So G is closed under multiplication.

 

b.

Soln:

Since all the complex numbers satisfy associative law under multiplication. So all the elements of G being complex numbers also satisfy associative law.

Ie. a * (b * c) = (a * b) * c, for all a,b,c ԑ G.

 

c.

Soln:

For all a ԑ G, a * 1 = 1 * a = a.

So, 1 is the multiplication identity in G.

Since, 1 * 1 = 1

(-1) * (-1) = 1

i * (-i) = -i2 = 1

and (-i) * I = -i2 = 1

So, the inverse of 1, -1, i and – I and 1, - 1, - i and i , respectively. So ,every element of G possesses an inverse element in G. Hence, identity element and inverse exists.

 

d.

Soln:

Yes, G forms a group under multiplication as (G,x) is closed, associative, and the identity and inverse exist in G.

 


Question no. 6

Soln:

Let a,b ԑ Z.

Since, a,b ԑ Z à a + b ԑ Z.

So, closure property is satisfied.

If a,b,c ԑ Z, then

a + (b + c) = a + b + c ԑ Z.

Also, (a + b) + c = a + b + c ԑ Z.

So, a + (b + c) = (a + b) + c.

Hence, associative property is satisfied. 0 is an integer and

0 + a = a + 0 = a ԑ Z.

0 is an identity element.

Also, if a ԑ Z then – a ԑ Z.

Or, a + (- a) = (-a) + a = 0.

So, - a is the inverse element of a. Above relations are true for all elements of Z.

Hence, the set of integers Z forms a group under addition.

 


Question no. 7

Soln:

*

a

b

c

a

a

b

c

b

b

c

a

c

c

a

b

 

From the table, we see that the operation defined on any two elements of G gives and element of G itself.

So, G is closed under the operation *.

a * (b * c) = a * a = a

(a * b) * c = b * c = a

So, a * (b * c) = (a *b) * c

So, * satisfies associative property.

Since, a * a = a, a * b = b * a= b

And a * c = c * a = c, so a is an identity element.

a * a = a, so a is the inverse of a,b * c = c * b = a, so b and c are the inverse elements of c and  respectively,

So, (S,*) forms a group.

 


Question no. 8

Soln:

Composition table for G under the addition modulo r (+4) is presented below.

+4

0

1

2

3

0

0

1

2

3

1

1

2

3

0

2

2

3

0

1

3

3

0

1

2

 

From the table, we see that sum of any two elements of G modulo 4 is an element of G. So, +4 satisfies closure property.

Again, 1 +4 (2 +4 3) = 1 +4 1 = 2.

And (1 +4 2) +4 3 = 3 +4 3 = 2.

This result is true for all elements of G. Hence, +4 satisfies associative property.

From the second row and second column of above table, 0 is the identity element.

Form the second row and second column of above table, 0 is the identity elemnt.

Since, 0 +4 0 = 0, 1 +4 3 = 3 +4 1 = 0

And 2 +4 2 = 0

So, the inverse elements of 0,1,2 and 3 are 0,3,2 and 1 respectively.

So, G forms a group under addition modulo 4.

 


Question no. 9

Soln:

Since a is an identity element, so,

a * a = a,  a * b = b * a = a.

So, also a * a = a, a is the inverse of a.

And b * b = a so that b is the inverse of b.

Now the required composite table is given below.

*

a

b

a

a

b

b

b

a



Elementary Group Theory - Exercise 3.3 : Class 12 Math PDF



Read:


Elementary Group Theory - Exercise 3.1 : Class 12 Math

Elementary Group Theory - Exercise 3.2 : Class 12 Math

Elementary Group Theory - Exercise 3.4 : Class 12 Math

Getting Info...

About the Author

Iswori Rimal is the author of iswori.com.np, a popular education platform in Nepal. Iswori helps students in their SEE, Class 11 and Class 12 studies with Complete Notes, important questions and other study materials.

Post a Comment

AdBlock Detected!
We have detected that you are using adblocking plugin in your browser.
The revenue we earn by the advertisements is used to manage this website, we request you to whitelist our website in your adblocking plugin.
Site is Blocked
Sorry! This site is not available in your country.