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Converse, inverse, contrapositive of prime statement
Short answer
Original: If a positive integer is prime then it has no divisors other than 1 and itself (p→q). Converse (q→p): If it has no such divisors then it is prime. Inverse (¬p→¬q): If not prime then it has another divisor. Contrapositive (¬q→¬p): If it has another divisor then it is not prime.
State the converse, contrapositive and inverse of the statement, "A positive integer is prime only if it has no divisors other than 1 and itself".
Let p: "a positive integer is prime" and q: "it has no divisors other than 1 and itself".
"p only if q" means p → q. So the given statement is: If a positive integer is prime, then it has no divisors other than 1 and itself.
| Form | Symbolic | Statement |
|---|---|---|
| Converse | q → p | If a positive integer has no divisors other than 1 and itself, then it is prime. |
| Inverse | ¬p → ¬q | If a positive integer is not prime, then it has a divisor other than 1 and itself. |
| Contrapositive | ¬q → ¬p | If a positive integer has a divisor other than 1 and itself, then it is not prime. |
The contrapositive is logically equivalent to the original statement. The converse and the inverse are equivalent to each other, but not to the original statement in general.
✓ Answer checked by Claude Opus 5.5 (high effort)
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