### 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.

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.

iii) Multiplication: Multiplying two whole numbers result in another whole number. Hence, whole numbers under multiplication are closed.

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.

Integers:

ii) Subtraction: Subtracting two integers result in another integer. Hence, integers under subtraction is closed.

iii) Multiplication: Multiplying two integers result in another integer. Hence, integers under multiplication is closed.

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.

Rational Numbers:

i) Addition: Adding two rational numbers result in another rational number. Hence, rational numbers under addition is closed.

$\frac{a}{b}+\frac{c}{d}$

ii) Subtraction: Subtracting two rational numbers result in another rational number. Hence, rational numbers under subtraction are closed.

$\frac{a}{b}-\frac{c}{d}$

$\frac{8}{5}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\frac{\left(-2\right)}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{8-\left(-2\right)}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{10}{5}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\phantom{\rule{0.147em}{0ex}}\mathit{or}\phantom{\rule{0.147em}{0ex}}2/1$

iii) Multiplication: Multiplying two rational numbers result in another rational number. Hence, rational numbers under multiplication is closed.

$\frac{a}{b}×\frac{c}{d}$

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.

$\frac{a}{b}÷\frac{c}{d}$