# State whether the following statement is true or false. Justify your answer.

The set of all integers is contained in the set of all rational numbers.

\[{\text{(A)}}\]True

\[{\text{(B)}}\]False

Answer

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Hint:- Use Venn’s Diagram.

As we know that,

A rational number is of the form \[\dfrac{p}{q}\] where q is not equal to zero.

So, some rational numbers will be \[\dfrac{{ - 5}}{3},\dfrac{2}{5},\dfrac{1}{7},\dfrac{5}{1},\dfrac{{ - 3}}{1}\]etc.

And, integer is a whole number (not a fractional number) that can be positive, negative, or zero.

As if we take, \[q = 1\] then every rational number will become integer.

And we know any set X is contained in any other set Y if all elements of X also belong to Y.

So, every integer number is written as a rational number with q=1.

Hence, the statement is true. The set of all integers is contained in the set of all rational numbers.

Note:- Whenever we came up with this type of problem then we should go with the

definition of rational numbers, integers, whole numbers and non-rational numbers.

It will be the easiest and efficient way to prove the result.

As we know that,

A rational number is of the form \[\dfrac{p}{q}\] where q is not equal to zero.

So, some rational numbers will be \[\dfrac{{ - 5}}{3},\dfrac{2}{5},\dfrac{1}{7},\dfrac{5}{1},\dfrac{{ - 3}}{1}\]etc.

And, integer is a whole number (not a fractional number) that can be positive, negative, or zero.

As if we take, \[q = 1\] then every rational number will become integer.

And we know any set X is contained in any other set Y if all elements of X also belong to Y.

So, every integer number is written as a rational number with q=1.

Hence, the statement is true. The set of all integers is contained in the set of all rational numbers.

Note:- Whenever we came up with this type of problem then we should go with the

definition of rational numbers, integers, whole numbers and non-rational numbers.

It will be the easiest and efficient way to prove the result.

Last updated date: 21st Sep 2023

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