Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $$p \Rightarrow (q \vee r)$$ is:

We are given the statement $$ p \Rightarrow (q \vee r) $$ and need to find which option is logically equivalent to it. Remember that $$ a \Rightarrow b $$ is equivalent to $$ \sim a \vee b $$. So, let's rewrite the given statement:

$$ p \Rightarrow (q \vee r) = \sim p \vee (q \vee r) $$

Since disjunction (OR) is associative, we can write this as:

$$ \sim p \vee q \vee r $$

Now, we'll check each option by converting them into equivalent forms using the same implication rule.

Option A: $$ (p \vee q) \Rightarrow r $$

Rewrite the implication:

$$ (p \vee q) \Rightarrow r = \sim (p \vee q) \vee r $$

Apply De Morgan's law to $$ \sim (p \vee q) $$:

$$ (\sim p \wedge \sim q) \vee r $$

This is not the same as $$ \sim p \vee q \vee r $$ because it has a conjunction ($$\wedge$$) and different terms. For example, if $$ p $$ is false, $$ q $$ is false, and $$ r $$ is false, the original statement is true (since $$ \sim p $$ is true), but this expression becomes $$ (\text{true} \wedge \text{true}) \vee \text{false} = \text{true} \vee \text{false} = \text{true} $$. However, if $$ p $$ is true, $$ q $$ is false, and $$ r $$ is false, the original is false (true implies false), but this expression is $$ (\text{false} \wedge \text{true}) \vee \text{false} = \text{false} \vee \text{false} = \text{false} $$, which matches. But let's test another case: $$ p $$ false, $$ q $$ true, $$ r $$ false. Original: false implies (true or false) = false implies true = true. Option A: $$ (\text{false} \vee \text{true}) \Rightarrow \text{false} = \text{true} \Rightarrow \text{false} = \text{false} $$. Not the same. So, not equivalent.

Option B: $$ (p \Rightarrow q) \vee (p \Rightarrow r) $$

Rewrite each implication:

$$ p \Rightarrow q = \sim p \vee q $$

$$ p \Rightarrow r = \sim p \vee r $$

So the expression becomes:

$$ (\sim p \vee q) \vee (\sim p \vee r) $$

Since disjunction is associative and commutative, rearrange:

$$ \sim p \vee \sim p \vee q \vee r $$

Simplify ($$ \sim p \vee \sim p = \sim p $$):

$$ \sim p \vee q \vee r $$

This matches the original expression exactly. So, option B is equivalent.

Option C: $$ (p \Rightarrow \sim q) \wedge (p \Rightarrow r) $$

Rewrite each implication:

$$ p \Rightarrow \sim q = \sim p \vee \sim q $$

$$ p \Rightarrow r = \sim p \vee r $$

So the expression is:

$$ (\sim p \vee \sim q) \wedge (\sim p \vee r) $$

Factor out $$ \sim p $$ using distribution:

$$ \sim p \vee (\sim q \wedge r) $$

This is not the same as $$ \sim p \vee q \vee r $$. For example, if $$ p $$ is true, $$ q $$ is true, and $$ r $$ is false, the original statement is true (true implies (true or false) = true implies true = true), but this expression is $$ \text{false} \vee (\text{false} \wedge \text{false}) = \text{false} \vee \text{false} = \text{false} $$. Not equivalent.

Option D: $$ (p \Rightarrow q) \wedge (p \Rightarrow \sim r) $$

Rewrite each implication:

$$ p \Rightarrow q = \sim p \vee q $$

$$ p \Rightarrow \sim r = \sim p \vee \sim r $$

So the expression is:

$$ (\sim p \vee q) \wedge (\sim p \vee \sim r) $$

Factor out $$ \sim p $$:

$$ \sim p \vee (q \wedge \sim r) $$

This is not the same as $$ \sim p \vee q \vee r $$. For example, if $$ p $$ is true, $$ q $$ is false, and $$ r $$ is true, the original statement is true (true implies (false or true) = true implies true = true), but this expression is $$ \text{false} \vee (\text{false} \wedge \text{false}) = \text{false} \vee \text{false} = \text{false} $$. Not equivalent.

Therefore, only option B is logically equivalent to the given statement. Hence, the correct answer is Option B.