Let $X$ be a non-empty set and $P(X)$ be the set of all subsets of $X$. On $P(X)$, define two operations $\ast$ and $\Delta$ as follows: for $A,B\in P(X)$,

• $A\ast B=A\cap B$ (Intersection of sets)
• $A\Delta B=(A\cup B)\setminus (A\cap B)$ (Symmetric difference of sets)

Which of the following statements is true?

Correct option is C

1. Operation $\ast$ (Intersection) on Power Set $P(X)$:

The operation $\ast$ is defined as:

$$A\ast B=A\cap B$$

This means that $A\ast B$ represents the intersection of sets $A$ and $B$.

• Identity Element: There must exist an identity element $e$ such that for every set $A$, we have:

  $$A\ast e=A\text{and}e\ast A=A$$
  • No identity element: For intersection, the identity element should be the set $X$, but $X\cap A=A$ holds for any set $A$.​​
  • The problem is that the inverse of each element does not exist, meaning there is no set such that A∗A′=XA \ast A' = XA∗A′A*A'A∗A′​=X,
  • because there is no set that can "reverse" the intersection. Thus, no inverse exists.

Since the operation $\ast$ on $P(X)$ fails the condition of having inverses for each element, $P(X),\ast$ is not a group.

2. Operation $\Delta$ (Symmetric Difference) on Power Set $P(X)$:

The operation $\Delta$ is defined as:

$$A\Delta B=(A\cup B)\setminus (A\cap B)$$

This operation represents the symmetric difference between two sets, which is the set of elements that are in either of the sets, but not both.

To form a group under $\Delta$, the following conditions must be satisfied:

• Closure: For any two subsets $A$ and $B$ of $X$, the symmetric difference $A\Delta B$ must also be a subset of $X$.

  • Closure holds: The symmetric difference of two subsets is always a subset of $X$, so closure is satisfied.

• Associativity: For all subsets $A,B,C$ in $P(X)$, the operation must satisfy:

  $$(A\Delta B)\Delta C=A\Delta (B\Delta C)$$
  • Associativity holds: Symmetric difference is associative.

• Identity Element: There must be an identity element $e$ such that:

  $$A\Delta e=A\text{and}e\Delta A=A$$
  • Identity is the empty set: The empty set $\varnothing$ acts as the identity element for symmetric difference because $A\Delta \varnothing =A$ for any set $A$.

• Inverse Element: For every element $A$, there must be an element A′A' A′A'A′ such that  AΔA′=∅

  
  
  • Inverse exists: The inverse of a set under symmetric difference is the set itself. Thus, for any set $A$, $A\Delta A=\varnothing$, satisfying the inverse condition.

Since the operation $\Delta$ satisfies closure, associativity, has an identity element (the empty set $\varnothing$), and every element has an inverse, $P(X),\Delta$ forms a group.

Conclusion:

• $P(X),\ast$ is not a group because the inverse of each element does not exist.
• $P(X),\Delta$ is a group with identity $\varnothing$.