What is Closure Property in Mathematics: Definition, Formula and Examples

Number System

Explanation

Example

Closure Under Addition

Natural Numbers (ℕ)

Adding any two natural numbers always gives another natural number. 

∀ a, b ∈ ℕ ⇒ a + b ∈ ℕ 

6 + 9 = 15 and 15 is a natural number.

Closed

Whole Numbers (W)

Whole numbers include zero along with the natural numbers, and adding them still stays inside the set.

∀ a, b ∈ W ⇒ a + b ∈ W 

0 + 23 = 23, a whole number.

Closed

Integers (ℤ )

Even when negative numbers are involved, the sum of two integers is always an integer.

 ∀ a, b ∈ ℤ ⇒ a + b ∈ ℤ 

-12 + 5 = -7, still an integer.

Closed

Rational Numbers (ℚ)

Adding two fractions with different denominators still produces a rational number

 after finding a common denominator.

 ∀ a, b ∈ ℚ ⇒ a + b ∈ ℚ 

 $\frac{2}{5} + \frac{1}{6} = \frac{12}{30} + \frac{5}{30} = \frac{17}{30}$, a rational number.

Closed

Number System

Explanation

Example / Counterexample

Closure Under Subtraction

Natural Numbers (ℕ)

Subtracting a bigger natural number from a smaller one gives a negative answer, which is not a natural number.

4 - 10 = -6, which is not a natural number.

Not Closed

Whole Numbers (W)

Whole numbers do not include negative numbers, so subtraction may produce a value outside the set.

3 - 8 = -5, which is not a whole number.

Not Closed

Integers (ℤ )

Since integers include negative numbers, subtracting any two integers always gives an integer.

 ∀ a, b ∈ ℤ ⇒ a - b ∈ ℤ 

-9 - 6 = -15 and 6 - (-9) = 15, both are integers.

Closed

Rational Numbers (ℚ)

Subtracting one rational number from another always gives a rational number.

 ∀ a, b ∈ ℚ ⇒ a - b ∈ ℚ 

 $\frac{3}{4} - \frac{1}{2} = \frac{3}{4} - \frac{2}{4} = \frac{1}{4}$, a rational number.

Closed

Number System

Explanation

Example

Closure Under Multiplication

Natural Numbers (ℕ)

Multiplying two natural numbers always gives another natural number.

∀ a, b ∈ ℕ ⇒ a × b ∈ ℕ

 $8 \times 7 = 56$, a natural number.

Closed

Whole Numbers(W)

This holds even when zero is one of the numbers involved.

∀ a, b ∈ W ⇒ a × b ∈ W

 $0 \times 15 = 0$, still a whole number.

Closed

Integers (ℤ )

Multiplying two negative integers gives a positive integer, which still belongs to the set.

∀ a, b ∈ ℤ ⇒ a × b ∈ ℤ

 $(-6) \times (-4) = 24$, an integer.

Closed

Rational Numbers(ℚ)

Multiplying two rational numbers always gives another rational number.

∀ a, b ∈ ℚ ⇒ a × b ∈ ℚ

 $\frac{2}{3} \times \frac{5}{9} = \frac{10}{27}$, a rational number.

Closed

Number System

Explanation

Counterexample

Closure Under Division

Natural Numbers (ℕ)

Dividing one natural number by another does not always give a natural number.

 $9 \div 4 = \frac{9}{4},$ which is not a natural number.

Not Closed

Whole Numbers (W)

Division may produce a fraction, which is not a whole number.

 $7 \div 3 = \frac{7}{3},$ which is not a whole number.

Not Closed

Integers (ℤ )

Dividing one integer by another often results in a fraction.

 $10 \div 4 = \frac{5}{2},$ which is not an integer.

Not Closed

Rational Numbers (ℚ)

Rational numbers are closed under division only when the divisor is non-zero. Division by zero is undefined.

 $7 \div 0$ is undefined, so it is not a rational number.

Not Closed