Consider three statements
p : Aman will come today
q : Aditi will be at her home
r : They will go to party.
Then, the statement `(q^^~r) rArrp` is logically equivalent to

Options

A. If Aman will not come today then they will go to party.
B. If Aman will not come today then they will not go to party.
C. Aman will come today or Aditi will not be at her home or they will go to party.
D. Aman will come today or Aditi will go to party.

Answer

To solve the problem, we need to simplify the logical statement \( (q \land \neg r) \implies p \) and find its logically equivalent form.

### Step 1: Understand the statement
The given statement is \( (q \land \neg r) \implies p \). Here:
- \( p \): Aman will come today
- \( q \): Aditi will be at her home
- \( r \): They will go to the party

### Step 2: Apply the implication equivalence
Recall that an implication \( A \implies B \) can be rewritten as \( \neg A \lor B \). Thus, we can rewrite our statement:
\[ (q \land \neg r) \implies p \equiv \neg(q \land \neg r) \lor p \]

### Step 3: Apply De Morgan's Law
Using De Morgan's Law, we can simplify \( \neg(q \land \neg r) \):
\[ \neg(q \land \neg r) \equiv \neg q \lor \neg(\neg r) \equiv \neg q \lor r \]
So, substituting this back into our expression gives:
\[ \neg(q \land \neg r) \lor p \equiv (\neg q \lor r) \lor p \]

### Step 4: Rearranging the expression
We can rearrange the expression using the associative and commutative properties of disjunction:
\[ \neg q \lor r \lor p \equiv p \lor r \lor \neg q \]

### Step 5: Final expression
Thus, the logically equivalent expression to \( (q \land \neg r) \implies p \) is:
\[ p \lor r \lor \neg q \]

### Conclusion
The final answer is \( p \lor r \lor \neg q \), which can be interpreted as:
- Aman will come today, or
- They will go to the party, or
- Aditi will not be at her home.