Group Theory Discrete Mathematics


#1

Qn) Z is a set of Integers, (Z,*) is a group with a*b= a+b+1, a,b ∈ Z. Then the inverse of a is
(i) -a
(ii) a+1
(iii) -2-a
(iv)None


#2

First of all we need to find the identity element.
a*e=a, we need to find ‘e’. ----> equation(1)

We know a*b=a+b+1
Now a*e=a+e+1 ---->equation(2)

Now equating equation(1) & (2),
a+e+1=a
e=-1

Now let ‘b’ be the inverse of ‘a’
We know a*b=e
a+b+1=-1
b=-2-a

So option (iii) is correct.


#3

a*b = a + b + 1.

for inverse we need to find identity element.
identity is a.e = e.a = a

so e = a -1 .
since a - 1 + 1 = a.
so e = -1.

inverse is a.inverse = e.

so -a - 2

a - a - 2 + 1 = - 1.

so e is unique and inverse depends on element or e can be different for different a too in other groups ?? @Ruturaj


#4

There is an Important property of the group. If Inverse exists then it is unique. There is a simple prove for this also.

a*(a^-1)=e -----> eqn(i)

Let ‘b’ be another inverse.Then a*b=e ---->eqn(ii)
Now let’s equate both the equations.

a*(a^-1)=a*b

applying left cancellation rule,

we have a^(-1) = b

So this implies that the inverse is unique.


#5

yaa unique in the sense for a particular element,but e is same irrespective of element or may be im missing something because in the question above ,inverse was -a - 2,so if a = 1,inverse id -3,a= 3 inverse is -5


#6

Inverse of a=1 is -1-2= -3
Inverse of a = -3 is -(-3)-2 = 3-2 = 1


#7

so inverse depends on which element im choosing and that will be unique,
How about identity element,does identity element also depends on which element we are doing the group operation


#8

No, identity element is same for all elements in the group.
ab = e when a,b are inverse to each other.
c
d = e when c,d are inverse to each other .
here e is identity element.