State whether the following statements are true or false. Justify your answers.
All hexagons are polygons.
True. A hexagon is a closed figure with six sides. A polygon is defined as a closed figure with three or more straight sides. Since a hexagon satisfies this definition, all hexagons are polygons.
Some polygons are pentagons.
True. A pentagon is a polygon with five sides. Since pentagons are a specific type of polygon, some polygons (namely pentagons) are pentagons.
Not all even numbers are divisible by 2.
False. By definition, even numbers are integers that are divisible by 2. Therefore, all even numbers are divisible by 2, making the statement false.
Some real numbers are irrational.
True. As established in Example 4 of the context, √p is irrational for all primes p. This confirms that irrational numbers exist and are real numbers.
Not all real numbers are rational.
True. Since irrational numbers exist (as shown in Example 4), and irrational numbers are real numbers that cannot be expressed as rational numbers, not all real numbers are rational.
Explanation
The context from Exercise A1.1 Question 2 lists these statements to be evaluated. The reasoning follows the pattern shown in Example 2, where statements about mathematical classifications (like 'All equilateral triangles are isosceles' and 'Some rational numbers are integers') are justified using definitions. For parts (iv) and (v), Example 4 explicitly establishes that irrational numbers exist (√p is irrational for primes), providing direct support from the context.