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 analyze the logical statement given: \((q \land \neg r) \implies p\). We will break it down step by step.

### Step 1: Understand the Components
We have three statements:
- \(p\): Aman will come today.
- \(q\): Aditi will be at her home.
- \(r\): They will go to the party.

The expression we need to analyze is \((q \land \neg r) \implies p\).

### Step 2: Rewrite the Implication
Recall that an implication \(A \implies B\) can be rewritten using logical equivalence as \(\neg A \lor B\). Thus, we can rewrite our expression:

\[ (q \land \neg r) \implies p \equiv \neg(q \land \neg r) \lor p \]

### Step 3: Apply De Morgan's Law
Using De Morgan's Law, we can simplify \(\neg(q \land \neg r)\):

\[ \neg(q \land \neg r) \equiv \neg q \lor \neg(\neg r) \equiv \neg q \lor r \]

### Step 4: Substitute Back into the Expression
Now substitute this back into our rewritten implication:

\[ \neg(q \land \neg r) \lor p \equiv (\neg q \lor r) \lor p \]

### Step 5: Rearranging the Expression
Using the associative property of logical disjunction, we can rearrange this expression:

\[ \neg q \lor r \lor p \]

### Conclusion
Thus, the statement \((q \land \neg r) \implies p\) is logically equivalent to:

\[ \neg q \lor r \lor p \]