For better understanding let start the question by taking set X = {1,2} and P(X) denotes the power set of X. So,
P(X) = { $\phi$, {1}, {2}, {1,2} }.
A binary operation $\Delta$ is defined on P(X) such that :
A $\Delta$ B = (A-B) $\cup$ (B-A)
Simply saying $\Delta$ is symmetric difference.
And, H = ({ $\phi$, {1}, {2}, {1,2} }, $\Delta$).
Let's take a look of on the closure property, Associativity, Identity & Inverse for H.
Closure Property : H is said to be satisfying closure property iff we take any two element from
{ $\phi$, {1}, {2}, {1,2} } the result will always belongs to this itself.
Example : Let A = {1} and B = $\phi$
Then, (A-B) $\cup$ (B-A) = ({1}-$\phi$) $\cup$ ($\phi$ - {1}) = {1} $\cup$ $\phi$ = {1} $\epsilon$ P(X).
If we try to take any two elements from P(X) then we will observe that result will belongs to P(X).
So, H satisfies Closure Property.
Associativity : (A $\Delta$ B) $\Delta$ C = A $\Delta$ (B $\Delta$ C).
Let's take A = {1}, B = $\phi$ & C = {2}
LHS : (A $\Delta$ B) $\Delta$ C
= {1} $\Delta$ C = {1} $\Delta$ {2}
= ({1} -{2}) $\cup$ ({2}-{1}) = {1,2}
RHS : A $\Delta$ (B $\Delta$ C)
= A $\Delta$ ( $\phi$ $\Delta$ {2}) = A $\Delta$ ( ($\phi$ - {2}) $\cup$ ({2} - $\phi$) )
= A $\Delta$ {2} = {1} $\Delta$ {2}
= {1,2}
Hence, LHS = RHS.
So, H satisfies Associativity.
Identity Property : An element 'e' is said to be identity element iff
A $\Delta$ E = E $\Delta$ A = A.
It can easily be observed that for this binary operation $\Delta$, $\phi$ is the identity element. E = $\phi$
Let's have a look How?
A $\Delta$ E = (A - $\phi$) $\cup$ ($\phi$ - A) = A $\cup$ $\phi$ = A
E $\Delta$ A = ($\phi$ - A) $\cup$ (A - $\phi$) = $\phi$ $\cup$ A = A
Hence, E = $\phi$ is the identity element.
So, H have Identity Element.
Inverse : Let, $A^{-1}$ be the inverse of A. An element is said to be inverse if
A $\Delta$ $A^{-1}$ = $A^{-1}$ $\Delta$ A = E
Let's 'T' be any element from P(X) and try to find out it's inverse
T $\Delta$ $T^{-1}$ = $\phi$
(T - $T^{-1}$) $\cup$ ($T^{-1}$ - T) = $\phi$
This equation will only holds good when $T^{-1}$ = T, means every element should be it's own inverse.
So, H have Inverse Element.
As, H satisfies Closure Property, Associativity, Indetity Element & Inverse. Then H is a group.
With this conclusion Option A is accepted and Option B is Rejected.
From conclusion on inverse property we can conclude that Option C is Rejected and
Option D is accepted.
So, correct comination of Answer : A, D.