2. Closure property

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.

 

$\begin{array}{l}5-3 = 2 (\mathit{whole})\\ \\ 0 - 6 =\; -6 (\mathit{not} \mathit{whole})\end{array}$

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

 

$\begin{array}{l}5 \times 3 = 15\\ \\ 2 \times 0 = 0\end{array}$

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.

i) Addition: Adding two integers result in another integer. Hence, integers under addition are closed.

 

$\begin{array}{l}(-2)\; +3 = 1\\ \\ (-7)\; +\; (-5)\; =\; -12\end{array}$

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

 

$\begin{array}{l}5-(-3)\; = 8\\ \\ (-3)\; - 6 =\; -9\end{array}$

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

 

$\begin{array}{l}5 \times (-3)\; =\; -15\\ \\ (-2) \times (-5)\; = 10\end{array}$

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.

 

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

 

$\frac{a}{b}+\frac{c}{d}$ 

 

$\frac{8}{5}+\frac{ (-2)}{5}=\frac{8+\left(-2\right)}{5}=\frac{6}{5}$

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

 

 

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

 

 

$\frac{8}{5}\times \frac{(-2)}{5} =\frac{8\times (-2)}{25} =\frac{-16}{25}$

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.