Question

Is subtraction Associative in rational numbers​

Answer 1

no

Step-by-step explanation:

First, let’s clarify what ‘associative’ means:

Associativity means you can perform an operation regardless of the grouping of numbers to achieve the same result, i.e. a+(b+c)=(a+b)+c

e.g.

4+(6+1)=4+(7)=11

(4+6)+1=(10)+1=11

For integers, addition is associative.

Is subtraction associative over rational numbers?

Let’s see. A rational number is one that can be written in the form pq:p,q∈Z

So, the question is:

Is it always true that ab−(cd−fg)=(ab−cd)−fg ?

We could formally prove that the answer is no, but we don’t need to since we can provide a simple counter-example which shows that the answer is no.

Counter-example:

Consider 123−(93−63)=(123−93)−63

LHS=

123−(93−63)

=123−(33)

=93

=3

RHS=

(123−93)−63

=(33)−63

=−33

=−1

Since 3≠−1 , LHS ≠ RHS

Thus, subtraction over rational numbers is not associative.