Which of the following sets are closed under division?
1) integers 2) irrational numbers 3) whole numbers

Solution:

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.

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 an 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.

Thus, Integers, Irrational numbers, and Whole numbers - None of these sets are closed under division.

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Which of the following sets are closed under division?
1) integers 2) irrational numbers 3) whole numbers

Summary:

Integers, Irrational numbers, and Whole numbers none of these sets are closed under division.