Question

Для данной импликации «Если натуральное число делится на 2 и 5, то оно делится на 10» сформулируйте обратную (В→А), противоположную (¬A→¬B), противоположную к обратной (¬B→¬A). Какие из четырех импликаций истинные, а какие ложные?

Answer 1

Прямая: Если натуральное число делится на 2 и 5, то оно делится на 10 - истинная.

Обратная: Если натуральное число делится на 10, то оно делиться на 2 и 5 - истинная.

Противоположная: Если натуральное число не делится на 2 и 5, то оно не делится на 10 - истинная.

Противоположная к обратной: Если натуральное число не делится на 10, то оно не делиться на 2 и 5 - истинная.

Answer 2

Implication and its Contrapositive, Inverse, and Converse

The given implication is: "If a natural number is divisible by 2 and 5, then it is divisible by 10."

1. Contrapositive (¬B → ¬A): - The contrapositive of the given implication is: "If a natural number is not divisible by 10, then it is not divisible by 2 or 5." - This is derived by negating both the hypothesis and the conclusion of the original implication.

2. Inverse (¬A → ¬B): - The inverse of the given implication is: "If a natural number is not divisible by 2 and 5, then it is not divisible by 10." - This is derived by simply negating the hypothesis and the conclusion of the original implication.

3. Converse (B → A): - The converse of the given implication is: "If a natural number is divisible by 10, then it is divisible by 2 and 5." - This is obtained by interchanging the hypothesis and the conclusion of the original implication.

Truth Values of the Implications

Now, let's determine which of the four implications are true and which are false.

1. Given Implication (A → B): - The given implication is: "If a natural number is divisible by 2 and 5, then it is divisible by 10." - This implication is true. A natural number that is divisible by both 2 and 5 is indeed divisible by 10.

2. Contrapositive (¬B → ¬A): - The contrapositive is: "If a natural number is not divisible by 10, then it is not divisible by 2 or 5." - This implication is also true. If a number is not divisible by 10, it cannot be divisible by both 2 and 5.

3. Inverse (¬A → ¬B): - The inverse is: "If a natural number is not divisible by 2 and 5, then it is not divisible by 10." - This implication is false. A number not being divisible by 2 and 5 does not necessarily mean it is not divisible by 10.

4. Converse (B → A): - The converse is: "If a natural number is divisible by 10, then it is divisible by 2 and 5." - This implication is false. A number being divisible by 10 does not imply it is divisible by both 2 and 5.

In summary, the given implication and its contrapositive are true, while the inverse and the converse are false.