There are some properties of subtracting rational numbers like closure, commutative, associative and distributive.
The difference between any two rational numbers is always a rational number.
Hence Q is closed under subtraction.
If a/b and c/d are any two rational numbers, then (a/b) - (c/d) is also a rational number.
Example :
5/9 - 2/9 = 3/9 = 1/3 is a rational number.
Subtraction of two rational numbers is not commutative.
If a/b and c/d are any two rational numbers,
then (a/b) - (c/d) ≠ (c/d) - (a/b)
Example :
5/9 - 2/9 = 3/9 = 1/3
2/9 - 5/9 = -3/9 = -1/3
Hence, 5/9 - 2/9 ≠ 2/9 - 5/9
Therefore, Commutative property is not true for subtraction.
Subtraction of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers,
then a/b - (c/d - e/f) ≠ (a/b - c/d) - e/f
Example :
2/9 - (4/9 - 1/9) = 2/9 - 3/9 = -1/9
(2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9
Hence, 2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9
Therefore, Associative property is not true for subtraction.
Distributive property of multiplication over subtraction :
Multiplication of rational numbers is distributive over subtraction.
If a/b, c/d and e/f are any three rational numbers,
then a/b x (c/d - e/f) = a/b x c/d - a/b x e/f
Example :
1/3 x (2/5 - 1/5) = 1/3 x 1/5 = 1/15
1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5 = (2 - 1) / 15 = 1/15
Hence, 1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5
Therefore, multiplication is distributive over subtraction.
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