Final answer:
Positive integers are not closed under subtraction, rational numbers are closed under multiplication, negative real numbers are not closed under division, and irrational numbers are not closed under addition.
Step-by-step explanation:
Positive integers are not closed under subtraction because the result of subtracting two positive integers is not always a positive integer. For example, 3 - 5 results in -2, which is not a positive integer. Rational numbers are closed under multiplication because the product of any two rational numbers is always a rational number; for example, ⅝ x ⅙ equals ⅟. When it comes to negative real numbers, we say they are not closed under division because dividing one negative real number by another can result in a positive number (following the multiplication rule for signs), which is not a negative real number. Finally, irrational numbers are not closed under addition; the sum of two irrational numbers can sometimes result in a rational number, such as ∞ (∞ being the irrational number pi) and -∞, which would sum to 0, a rational number.