# PROPERTIES OF SUBTRACTION OF INTEGERS

In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z  =  {... - 2, - 1,0,1,2, ...}, is the set of all integers

Here, we are going to see the following the three properties of subtraction of integers.

(i)  Closure property

(ii)  Commutative property

(iii)  Associative property

## Closure Property of Subtraction of Integers

Observe the following examples:

(i)  12 - 5  =  7

(ii)  5 - 12  =  -7

(iii)  (-18) - (-13)  =  -18 + 13  =  -5

(iv)  (-13) - (-18)  =  -13 + 18  =  5

(v)  (-18) - 13  =  -18 - 13  =  -31

(vi)  18 - (-13)  =  18 + 13  =  31

From the above examples, it is clear that subtraction of any two integers is again an integer.

In general, for any two integers a and b, a - b is an integer.

Therefore, the set of integers is closed under subtraction.

## Commutative Property of Subtraction of Integers

Consider the integers 7 and 4. We see that

7 - 4 = 3

4 - 7 = - 3

Therefore,  7 - 4    4 - 7

In general, for any two integers a and b

a - b  ≠  b - a

Therefore, we conclude that subtraction is not commutative for integers.

## Associative Property of Subtraction of Integers

Consider the integers 7, 4 and 2

7 - (4 - 2)  =  7 - 2  =  5

(7 - 4) - 2  =  3 - 2  =  1

Therefore, 7 - (4 - 2)  ≠   (7 - 4) - 2

In general, for any three integers a , b and c

a - (b - c)  ≠  (a - b) - c.

Therefore, subtraction of integers is not associative.

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