The contrapositive of `(pvvq) to r` is

Options

A. `r to (pvvq)`
B. `~r to (pvvq)`
C. `~r to (~ p ^^~q)`
D. `p to (qvvr)`

Explanation

To find the contrapositive of the statement \( (p \lor q) \to r \), we can follow these steps:

### Step 1: Identify the original statement
The original statement is \( (p \lor q) \to r \). This means "If \( p \) or \( q \) is true, then \( r \) is true."

### Step 2: Understand the contrapositive
The contrapositive of a statement of the form \( A \to B \) is \( \neg B \to \neg A \). This means we need to negate both the conclusion and the premise of the original statement.

### Step 3: Negate the conclusion
In our case, the conclusion \( r \) needs to be negated. Thus, \( \neg r \) becomes "not \( r \)".

### Step 4: Negate the premise
The premise \( (p \lor q) \) also needs to be negated. The negation of \( (p \lor q) \) is \( \neg (p \lor q) \). According to De Morgan's laws, this can be rewritten as \( \neg p \land \neg q \), which means "not \( p \) and not \( q \)".

### Step 5: Write the contrapositive
Now we can write the contrapositive using the negated conclusion and premise:
\[ \neg r \to (\neg p \land \neg q) \]
This reads as "If \( r \) is not true, then both \( p \) and \( q \) are not true."

### Final Answer
The contrapositive of \( (p \lor q) \to r \) is:
\[ \neg r \to (\neg p \land \neg q) \]

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