The contra-positive of the statement, "If x is a prime number and x divides ab then x divides a or x divides b", can be symbolically represented using logical connec- tives, on appropriately defined statements p, q, r, s, as

Options

A. `(~r vee ~s) rarr (~p ^^ ~ q)`
B. `( r ^^ s) rarr (~p ^^ ~q)`
C. `(~r ^^ ~s) rarr (~p vee ~q) `
D. `( r vee s) rarr (~p vee ~ q ) `

Answer

To find the contrapositive of the statement "If x is a prime number and x divides ab, then x divides a or x divides b," we will follow these steps:

### Step 1: Define the Statements
We will define the statements as follows:
- Let \( p \): "x is a prime number"
- Let \( q \): "x divides ab"
- Let \( r \): "x divides a"
- Let \( s \): "x divides b"

### Step 2: Write the Original Statement in Symbolic Form
The original statement can be expressed symbolically as:
\[ p \land q \implies r \lor s \]
This reads as "If \( p \) and \( q \) are true, then \( r \) or \( s \) is true."

### Step 3: Find the Contrapositive
The contrapositive of a statement \( A \implies B \) is given by \( \neg B \implies \neg A \).

In our case:
- \( A \) is \( p \land q \)
- \( B \) is \( r \lor s \)

Thus, the contrapositive will be:
\[ \neg (r \lor s) \implies \neg (p \land q) \]

### Step 4: Apply De Morgan's Laws
Using De Morgan's laws, we can simplify the negations:
- \( \neg (r \lor s) \) becomes \( \neg r \land \neg s \)
- \( \neg (p \land q) \) becomes \( \neg p \lor \neg q \)

So, the contrapositive can be rewritten as:
\[ \neg r \land \neg s \implies \neg p \lor \neg q \]

### Final Result
The contrapositive of the original statement is:
\[ \neg r \land \neg s \implies \neg p \lor \neg q \]

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