### Theory:

A set of numbers is said to be commutative for a specific mathematical operation if the result obtained when changing the order of the operands does not change the result.
Whole Numbers:
i) Addition: Changing the order of operands in addition to whole numbers does not change the result. Hence, whole numbers under addition are commutative.

ii) Subtraction: Changing the order of operands in the subtraction of whole numbers changes the result. Hence, whole numbers under subtraction are not commutative.

iii) Multiplication: Changing the order of operands in the multiplication of whole numbers does not change the result. Hence, whole numbers under multiplication are commutative.

iv) Division: Changing the order of operands in the division of whole numbers changes the result. Hence, whole numbers under division are not commutative.

Integers:
i) Addition: Changing the order of operands in addition to integers, does not change the result. Hence integers under addition are commutative.

ii) Subtraction: Changing the order of operands in the subtraction of integers changes the result. Hence, integers under subtraction are not commutative.

iii) Multiplication: Changing the order of operands in the multiplication of integers does not change the result. Hence, integers under multiplication are commutative.

iv) Division: Changing the order of operands in the division of integers changes the result. Hence, integers under division are not commutative.

Rational Numbers:
i) Addition: Changing the order of operands in addition to rational numbers, does not change the result. Hence, rational numbers under addition are commutative.

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

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ii) Subtraction: Changing the order of operands in the subtraction of rational numbers changes the result. Hence, rational numbers under subtraction are not commutative.

$\frac{a}{b}-\frac{c}{d}\phantom{\rule{0.147em}{0ex}}\ne \phantom{\rule{0.147em}{0ex}}\frac{c}{d}-\frac{a}{b}$.

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iii) Multiplication: Changing the order of operands in the multiplication of rational numbers does not change the result. Hence, rational numbers under multiplication are commutative.

$\frac{a}{b}×\frac{c}{d}=\frac{c}{d}×\frac{a}{b}$.

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iv) Division: Changing the order of operands in the division of rational numbers changes the result. Hence, rational numbers under division are not commutative.

$\frac{a}{b}÷\frac{c}{d}\phantom{\rule{0.147em}{0ex}}\ne \phantom{\rule{0.147em}{0ex}}\frac{c}{d}÷\frac{a}{b}$.

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