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State True/False.

The rational numbers are always closed under subtraction.

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True

False

answer is A.

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Detailed Solution

Assume

a b

and

c d

be two rational numbers where

a, b, c and d

are integers and

b, d ≠ 0

. Then the difference between them is given by,

a b − c d = a d − b c b d

Since the product of integers is an integer, thus

a d, b c and b d

are integers.
The difference between two integers is also an integer.
So,

a d − b c

is an integer.
Since b and d are non-zero, so their product will also be non-zero. i.e.,

b d ≠ 0

. So,

a d − b c and b d

are integers and

b d ≠ 0

. Thus,

a d − b c b d

is a rational number i.e.,

a b − c d

is a rational numbers
As a result,

a b

and

c d

are two arbitrary rational numbers.
Hence, the difference between rational numbers is also a rational number. i.e., rational numbers are always closed under subtraction.
Thus, the given statement is true.
Therefore, option (1) is correct.

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