To determine which sets are closed under multiplication, we need to understand what it means for a set to be closed under a specific operation. A set is considered closed under multiplication if, whenever you take any two elements from that set and multiply them together, the result is also an element that is in that set.
Let’s evaluate some common number sets:
- Natural Numbers (N): The product of any two natural numbers is a natural number. For example, 2 x 3 = 6, which is in N. Thus, natural numbers are closed under multiplication.
- Whole Numbers (W): Similar to natural numbers, the product of any two whole numbers will also be a whole number. For example, 0 x 1 = 0, which is in W. Hence, whole numbers are closed under multiplication.
- Integers (Z): The product of any two integers is also an integer. For instance, -2 x 3 = -6, which belongs to Z. Therefore, integers are closed under multiplication.
- Rational Numbers (Q): The product of two rational numbers is a rational number. For example, (1/2) x (3/4) = 3/8, which is in Q. Hence, rational numbers are closed under multiplication.
- Real Numbers (R): The product of any two real numbers is a real number. For example, π x 2 = 2π, which is in R. Therefore, real numbers are closed under multiplication.
- Complex Numbers (C): The product of any two complex numbers is still a complex number. For example, (1 + i)(1 – i) = 2, which is in C. Thus, complex numbers are closed under multiplication.
In conclusion, all of the sets mentioned above—natural numbers, whole numbers, integers, rational numbers, real numbers, and complex numbers—are closed under multiplication. When selecting from a multiple-choice format, you would choose all these options, as they all meet the criteria for closure under multiplication.