To determine whether a set is closed under multiplication, we need to check if the product of any two elements in that set is also an element of the same set. Let’s consider a few common sets:
- Natural Numbers (N): The natural numbers are 1, 2, 3, and so on. When you multiply any two natural numbers, the result is always a natural number. For example, 2 x 3 = 6, which is still a natural number. Therefore, the set of natural numbers is closed under multiplication.
- Integers (Z): The integers include positive numbers, negative numbers, and zero. When you multiply any two integers, the result is also an integer. For example, -2 x 3 = -6, and 0 x 5 = 0. Hence, the set of integers is closed under multiplication.
- Rational Numbers (Q): Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. When multiplying two rational numbers (e.g., 1/2 x 3/4 = 3/8), the result is still a rational number. Therefore, the set of rational numbers is closed under multiplication.
- Real Numbers (R): The real numbers consist of all the rational and irrational numbers. When multiplying two real numbers, the product is also a real number (e.g., √2 x √3 = √6). Thus, the set of real numbers is closed under multiplication.
- Complex Numbers (C): The complex numbers include all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. The product of two complex numbers is also a complex number (e.g., (1 + i) x (2 + 3i) = -1 + 5i). Hence, the set of complex numbers is closed under multiplication.
In summary, the sets that are closed under multiplication include:
- Natural Numbers (N)
- Integers (Z)
- Rational Numbers (Q)
- Real Numbers (R)
- Complex Numbers (C)
To finalize, simply select all the aforementioned sets, as they all meet the criteria for closure under multiplication.