In mathematics, a set is closed under an operation when we perform that operation on members of the set, and we always get a set member. Thus, a set either has or lacks closure concerning a given operation. In general, a set that is closed under an operation or collection of functions is said to satisfy a closure property. Usually, a closure property is introduced as a hypothesis, traditionally called the axiom of closure.
Examples
Some of the important examples of a closure property include the following.
- The transitive closure of a set is an example of closure property in set theory.
- The transitive closure of a binary relation is an example of closure property in set theory.
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Closure property of addition
The best example of showing the closure property of addition is with the help of real numbers. Since the set of real numbers is closed under addition, we will get another real number when we add two real numbers. Here, there will be no possibility of ever getting anything (suppose complex number) other than another real number.
6 + 13 = 19
(1/3) + (5/2) = 17/6
6.24 + 7.5 = 13.74
5√5 – 2√5 = 3√5
Closure property for Integers
Closure property holds for addition, subtraction and multiplication of integers.
Closure property of integers under addition:
The sum of any two integers will always be an integer, i.e. if a and b are any two integers, a + b will be an integer.
Example: (-8) + 6 = 2
11 + 9 = 20
Closure property of integers under subtraction:
The difference between any two integers will always be an integer, i.e. if a and b are any two integers, a – b will be an integer.
Example: 19 – 6 = 13
-6 – (-3) = -3
Closure property of integers under multiplication:
Any two integers’ product will be an integer, i.e. if a and b are any two integers, ab will also be an integer.
Example: 3 × (-9) = -27
(–7) × (-9) = 63
Closure property of integers under division:
Division of integers doesn’t follow the closure property since the quotient of any two integers a and b, may or may not be an integer. Sometimes the quotient is undefined (when the divisor is 0).
Example: -16 ÷ 4 = -4 (an integer)
(−4) ÷ (−16) = 1/4 (not an integer)
Click here to know more about properties of integers.
Closure property of Rational Numbers
Closure property holds for addition, subtraction and multiplication of rational numbers.
Closure property of rational numbers under addition:
The sum of any two rational numbers will always be a rational number, i.e. if a and b are any two rational numbers, a + b will be a rational number.
Example: (5/6) + (2/3) = 3/2
-(1/2) + (1/4) = -1/4
Closure property of rational numbers under subtraction:
The difference between any two rational numbers will always be a rational number, i.e. if a and b are any two rational numbers, a – b will be a rational number.
Example: (7/8) – (3/8) = 1/2
(6/7) – (-3/7) = 9/7
Closure property of rational numbers under multiplication:
Closure property under multiplication states that any two rational numbers’ product will be a rational number, i.e. if a and b are any two rational numbers, ab will also be a rational number.
Example: (3/2) × (2/9) = 1/3
(-7/4) × (5/2) = -35/8
Closure property of rational numbers under division:
Division of rational numbers doesn’t follow the closure property since the quotient of any two rational numbers a and b, may or may not be a rational number. That means, it can be undefined when we take the value of b as 0.
Learn more about the properties of rational numbers here.
Closure property of Whole Numbers
Closure property holds for addition and multiplication of whole numbers.
Closure property of whole numbers under addition:
The sum of any two whole numbers will always be a whole number, i.e. if a and b are any two whole numbers, a + b will be a whole number.
Example: 12 + 0 = 12
9 + 7 = 16
Closure property of whole numbers under subtraction:
The difference between any two whole numbers may or may not be a whole number. Hence, the whole numbers are not closed under subtraction.
Example: 13 – 14 = -1 (not a whole number)
4 – 0 = 4 (whole number)
Closure property of whole numbers under multiplication:
Any two whole numbers’ product will be a whole number, i.e. if a and b are any two whole numbers, ab will also be a whole number.
Example: 4 × 6 = 24
0 × 7 = 0
Closure property of whole numbers under division:
Division of whole numbers doesn’t follow the closure property since the quotient of any two whole numbers a and b, may or may not be a whole number.
Example: 18 ÷ 4 = 9/2 (not a whole number)
22/2 = 11 (a whole number)
Closure property under Multiplication
The product of two real numbers is always a real number, that means real numbers are closed under multiplication. Thus, the closure property of multiplication holds for natural numbers, whole numbers, integers and rational numbers.
Examples:
8 × 0 = 0
(3/4) × (-1/2) = -3/8
√3 × √5 = √15
(-11) × (-3) = 33
Closure property under Division
The set of real numbers (includes natural, whole, integers and rational numbers) is not closed under division. Division by zero is the only case where closure property under division fails for real numbers. If we ignore this special case (division by 0), we can say that real numbers are closed under division.