Theory:

If a set of numbers is closed for a particular operation, then it is said to possess the closure property for that operation.
Whole Numbers:
i) Addition: Adding two whole numbers result in another whole number. Hence, whole numbers under addition are closed.
 
2+3 = 50 + 6 = 6

ii) Subtraction: Subtracting two whole numbers may result in a negative number which is not a whole number. Hence, whole numbers under subtraction are not closed.
 
53 = 2 (whole)0 − 6 = −6 (not whole)

iii) Multiplication: Multiplying two whole numbers result in another whole number. Hence, whole numbers under multiplication are closed.
 
5 × 3 = 152 × 0 = 0

iv) Division: Dividing two whole numbers may result in a fraction or a number with a decimal point which is not a whole number. Hence, whole numbers under subtraction are not closed.

4 ÷ 2 = 2 (whole)10 ÷ 4 = 10/4 = 2.5 (notwhole)
Integers:
i) Addition: Adding two integers result in another integer. Hence, integers under addition are closed.
 
(−2) +3 = 1(−7) + (−5) = −12

ii) Subtraction: Subtracting two integers result in another integer. Hence, integers under subtraction is closed.
 
5−(−3) = 8(−3) − 6 = −9

iii) Multiplication: Multiplying two integers result in another integer. Hence, integers under multiplication is closed.
 
5 × (−3) = −15(−2× (−5) = 10

iv) Division: Dividing two integers may result in a fraction or a number with a decimal point which is not an integer. Hence, integers under division are not closed.
 
4 ÷ (2) = 2 (Integer)(10) ÷ 4 10/4 2.5 (notaninteger)
Rational Numbers:
 
i) Addition: Adding two rational numbers result in another rational number. Hence, rational numbers under addition is closed.
 
ab+cd 
 
85+ (2)5=8+25=65

ii) Subtraction: Subtracting two rational numbers result in another rational number. Hence, rational numbers under subtraction are closed.
 
abcd
 
85(2)5=825=105=2or2/1


iii) Multiplication: Multiplying two rational numbers result in another rational number. Hence, rational numbers under multiplication is closed.
 
ab×cd
 
85× (2)5 =8×(2)25 =1625

iv) Division: Dividing two rational numbers may result in an undefined number with which is not a rational number. Hence, rational numbers under division is not closed.
 
ab÷cd
 
82÷ (2)2= 164= 82