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
converse
if n+1 is even, then n is prime (not necessarily so false)
if n = 9 then n+1 is even
but 9 = 3*3
contrapositive
if n+1 Is not even, then n is not prime (true if original statement is true, which it is)
but isnt 2 a prime # so 2+1=3 = odd
Good thinking.
I suspect that the original hypothesis should have stated greater than 2. That does not impact this problem because it is only asking for the converse and contrapositive.
E. Find a counterexample for the converse of the conditional statement. If u > -5, then u is a positive.
The given conditional statement is:
"If n is a prime number, then n+1 is an even number."
1. Converse:
The converse switches the order of the hypothesis (n is a prime number) and the conclusion (n+1 is an even number).
Converse: "If n+1 is an even number, then n is a prime number."
2. Contrapositive:
The contrapositive negates both the hypothesis and the conclusion of the original statement.
Contrapositive: "If n+1 is not an even number, then n is not a prime number."
Now, let's assess whether each of the three statements is true or false:
- The original conditional statement: "If n is a prime number, then n+1 is an even number."
Statement: True.
Explanation: This statement is true because if a number n is prime, then it cannot be divisible by any number except 1 and itself. In this case, if n is a prime number, n+1 will be even because it will always be divisible by 2.
- The converse: "If n+1 is an even number, then n is a prime number."
Statement: False.
Counterexample: Let n = 4. If we substitute n = 4 into the converse statement, "If 4+1 is an even number, then 4 is a prime number," we see that 5 is an odd number and not a prime number. Therefore, the converse statement is false.
- The contrapositive: "If n+1 is not an even number, then n is not a prime number."
Statement: True.
Explanation: This statement is true because if n+1 is not an even number (meaning it is an odd number), then n cannot be a prime number.