write the converse and contrapositive for the conditional statement below. decide whether each of the three statements is true or false. provide a counterexample for any false statement.
If n is a prime number, then n+1 is an even number
Any even number can be factored by 2, so by definition, n cannot be an even number. So what does that make n +1?
To determine the converse and contrapositive of a conditional statement, we need to switch the positions of the hypothesis and the conclusion:
Converse: If n+1 is an even number, then n is a prime number.
Contrapositive: If n+1 is not an even number, then n is not a prime number.
Now let's analyze the truth value of each statement:
Original Statement (If n is a prime number, then n+1 is an even number) - This statement is not universally true. It is possible to find counterexamples.
Counterexample: Let's consider n = 3. 3 is a prime number, but 3 + 1 = 4, which is an even number. Therefore, the original statement is false.
Converse (If n+1 is an even number, then n is a prime number) - This statement is also false. Counterexample: Let n = 4. 4 + 1 = 5, which is not a prime number.
Contrapositive (If n+1 is not an even number, then n is not a prime number) - This statement is true.
In summary:
- The original statement is false. (Counterexample: n = 3)
- The converse is false. (Counterexample: n = 4)
- The contrapositive is true.
Remember, a counterexample is a specific example that disproves a statement by showing that it does not hold in that case.