Adyar – Adambakkam – Pallavaram – Pammal – Chromepet – Selaiyur
+2 Mathematics
Group theory - Solutions
Q.No. 1
Solution :
Binary operation is defined as
x o y = x ; x, y  S
[ Meaning : (first element) o ( Second element) = First element ]
Commutative Axiom :
x o y = x ---- (1)
y o x = y ----- (2)
(1) ≠ (2)  Commutative axiom is not satisfied
Associative Axiom:
x o ( y o z ) = x o y = x ------(3)
(x o y ) o z = x o z = x ------(4)
(3)=(4)  Associataive axiom is satisfied
Conclusion : The binary operation o is Associative but not commutative
Q.No. 2
Solution :
N
=
Set of all natural numbers
Binary operation = x * y = Max ( x , y )
[ Example : 4 * 6 = 6 since 6 is highest ]
Closure axiom
:
for all x, y  N , x * y = Max(x,y)  N
Associative axiom
:
for all x, y , z  N
x * ( y * z ) = x * Max(y,z) = Max( x, y, z)  N
( x * y ) * z = Max(x,,y) * z = Max ( x,y,z)  N
Identity axion
:
here for any x  N
1 * x = x and x * 1 = x ( since 1 is the least natural number)
 identity element is 1
Identity element exists
Therefore N is a monoid under the operation *
Q. No 3
Solution :
G
= The set of all positive even integers
Under addition
Closure axiom
:
The sum of any two positive even integers is a positive even
Integer.  Closure axiom is true
Associative axiom
:
Usual addition is always associative
Identity axiom
:
Under addition, identity element is 0 , not belong to G
 Identity element does not exists
 The set G under addition is a semi-group but not a monoid
Under multiplication
Closure axiom
:
The product of any two positive even integers is also a positive
even integer.  Closure axiom is true
Associative axiom
:
Usual multiplication is always associative
Identity axiom
:
Under multiplication, identity element is 1, not belong to G
 Identity element does not exists
 The set G under multiplication is a semi-group but not a monoid
Q.No. 7
Solution
M = set of all complex numbers whose modulus is 1
Binary operation : Multiplication
Closure axiom
:
for all z1 , z2  M => |z1| = 1 and |z2| =1
| z1 . z2| = | z1| | z2| = (1) (1) =1
Closure axiom is true
Associative axiom
:
Usual multiplication is always associative
Identity axiom
:
Under multiplication , Identity element is 1  M
:
Since | 1 | = 1
for all z  M , such that | z| =1
Inverse axiom
Under multiplication , inverse of z = 1/z  M
Since |1/z| = |1| / |z| = 1/1 = 1
Inverse of each element exists.
M is a group with respect to multiplication
Q.No. 12
Solution
G
Binary operation
= { 2 n / n Z } i.e 2 to the power of any integer belong to the set G
= Mulitplication
Closure axiom
:
for all 2a . 2b  G , a , b  Z
2a . 2b = 2
a+ b
 G since a+b  Z  Closure axiom is true.
Associative axiom
:
Usual multiplication is always associative
Identity axiom
:
Under multiplication , identity element is 1 i.e 20  G , since 0  Z
Identity element exists
Inverse axiom
:
for all 2a  G, inverse of 2a is 2 –a  G since –a  Z
Also 2 a . 2 –a = 2 a –a = 2 0
Inverse of each element exists.
 G is a group under multiplication
Commutative axiom :
for all 2 a , 2 b  G such a , b  Z
2a . 2b = 2a+b = 2b+a = 2b . 2a
Commutative axiom is true
G is an abelian group under multiplication
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