Which of the following sets are closed under division?
1) integers 2) irrational numbers 3) whole numbers
The closure property of division states that if A, B are the two numbers that belong to a Set X then A ÷ B = C also belongs to set X.
Answer: Integers, Irrational numbers, and Whole numbers - None of these sets are closed under division.
Let us understand the concept of closure property.
Let a, b ∈ Z [ Z denoted the set of integers]
If a = 1, b = 0
1 / 0 ∉ Z
Thus, a ÷ b = a/ b ∉ Z
Thus, Integers are not closed under division.
Let's take and irrational number √2
√2 ∈ Q [Q is the set of irrational numbers]
Then, √2 ÷ √2 = 1 ∉ Q [Since, 1 is a rational number]
Thus, Irrational numbers are not closed under division.
Let a, b ∈ W [W is a set of whole numbers]
If a = 3, b = 0
3 / 0 ∉ W
Thus, a ÷ b = a/ b ∉ W.
Hence, Whole Numbers are not closed under division.