Which set is closed under division
The set of real numbers is closed under division, except for division by zero.
A set is closed under division if, for any two elements in the set, their quotient is also an element in the set.
There are several sets that are closed under division, such as the set of real numbers excluding zero (ℝ \ {0}), the set of rational numbers (ℚ), the set of complex numbers (ℂ), and the set of integers (ℤ).
However, it is important to note that the set of natural numbers (ℕ) is not closed under division, since division may yield a non-natural number.
To determine which set is closed under division, we need to understand what it means for a set to be closed under an operation. In this case, the operation is division.
A set, let's call it S, is closed under division if for any two elements a and b in S (a ≠ 0), the result of dividing a by b, denoted as a / b, is also an element of S.
Now, let's explore some common sets:
1. The set of natural numbers (N): This set does not fulfill the condition of closure under division, as division of two natural numbers may result in a non-natural number. For example, dividing 3 by 2 gives 1.5, which is not a natural number.
2. The set of integers (Z): This set is closed under division. If we take any two integers a and b (where b ≠ 0), the result of dividing a by b will also be an integer. For example, dividing 6 by 3 is 2, which is an integer.
3. The set of rational numbers (Q): This set is also closed under division. Any two rational numbers a and b (where b ≠ 0) divided will always result in another rational number. For example, dividing 3/4 by 1/2 is (3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 3/2, which is still a rational number.
4. The set of real numbers (R): This set is also closed under division. When dividing any two real numbers (except when dividing by 0), the result will always be a real number. For example, dividing 5.5 by 2.2 is 2.5, which is a real number.
In summary, the sets of integers (Z), rational numbers (Q), and real numbers (R) are all closed under division, while the set of natural numbers (N) is not.