The statement B`rArr((~A) vee B)` is equivalent to :

Options

A. A`rArr(AhArrB)`
B. B`rArr(ArArrB)`
C. `BrArr((~A)rArrB)`
D. `ArArr((~A)rArrB)`

Answer

To solve the logical statement \( B \rightarrow (\neg A \vee B) \), we will simplify it step by step.

### Step 1: Understand the implication
The statement \( B \rightarrow (\neg A \vee B) \) can be rewritten using the definition of implication. Recall that \( P \rightarrow Q \) is equivalent to \( \neg P \vee Q \).

So we can rewrite the statement as:
\[ \neg B \vee (\neg A \vee B) \]

### Step 2: Apply Associative Law
Using the associative law of disjunction, we can rearrange the terms:
\[ \neg B \vee \neg A \vee B \]

### Step 3: Apply the Disjunctive Law
Now, we can use the disjunctive law, which states that \( P \vee \neg P \) is always true (a tautology). Here, we have \( B \vee \neg B \):
\[ (\neg B \vee B) \vee \neg A \]

Since \( \neg B \vee B \) is a tautology (always true), we can simplify this to:
\[ \text{True} \vee \neg A \]

### Step 4: Final Simplification
Since anything ORed with true is true, we conclude that:
\[ \text{True} \]

Thus, the statement \( B \rightarrow (\neg A \vee B) \) is equivalent to a tautology.

### Conclusion
The statement \( B \rightarrow (\neg A \vee B) \) is equivalent to **True**.