Final answer:
A direct proof that the product of two rational numbers is rational assumes the commutative property of multiplication and the definition of rational numbers, which allows such a product to be expressed as the quotient of two integers.
Step-by-step explanation:
When proving the theorem that the product of two rational numbers (r and s) is a rational number, we begin by assuming basic arithmetic properties and definitions of rational numbers. Rational numbers are any numbers that can be expressed as a fraction of two integers, such that r = /a/b and s = /c/d, where a, b, c, and d are integers and b and d are not zero. For the multiplication of rational numbers, the assumption we rely on here is the commutative property of multiplication, which states that A'B = B'A, and the definition of multiplication for fractions as demonstrated by /a/b * /c/d = (a * c) / (b * d). Since integers are closed under multiplication, the product of two integers is also an integer, and therefore the product a * c and b * d are both integers, making (a * c) / (b * d) a rational number.
On the other hand, the division of two rational numbers does not always result in a rational number, as division by zero is undefined. Thus, option 4) is not always true and is not assumed in a direct proof.