Question

is subtraction associative in rational numbers? Explain with an example​

Answer 1

Step-by-step explanation:

Answer: No, subtraction is NOT associative in rational numbers.

Step-by-step explanation: We are asked to state whether subtraction is associative in rational numbers or not. To explain with an example.

Let a, b and c be three rational numbers. Then, according to the associative property, we must have

(a-b)-c=a-(b-c).(a−b)−c=a−(b−c).

The subtraction is NOT associative in rational numbers.

For example, let us consider that

a=\dfrac{1}{2},~b=\dfrac{3}{2},~c=\dfrac{5}{2}.a=

2

1

, b=

2

3

, c=

2

5

.

Then,

\begin{gathered}L.H.S.=(a-b)-c=\left(\dfrac{1}{2}-\dfrac{3}{2}\right)-\dfrac{5}{2}=-1-\dfrac{5}{2}=\dfrac{-2-5}{2}=-\dfrac{7}{2},\\\\\\R.H.S.=a-(b-c)=\dfrac{1}{2}-\left(\dfrac{3}{2}-\dfrac{5}{2}\right)=\dfrac{1}{2}-(-1)=\dfrac{1}{2}+1=\dfrac{3}{2}.\end{gathered}

L.H.S.=(a−b)−c=(

2

1

−

2

3

)−

2

5

=−1−

2

5

=

2

−2−5

=−

2

7

,

R.H.S.=a−(b−c)=

2

1

−(

2

3

−

2

5

)=

2

1

−(−1)=

2

1

+1=

2

3

.

So, L.H.S. ≠ R.H.S.

Hence, subtraction is NOT associative in rational numbers.