# Properties of Subtraction of Integers

Properties of  Subtraction of Integers: Closure Property, Commutative Property, Associative Property, Identity Property and Distributive Property are explained with examples.

(1) Closure under subtraction-

Subtraction of any two integers gives integer. It is known as integers are closed under subtraction.

Condition for closure property-

x  –  y = z

Here, x, y and z are integers.

Example

(a) 5-3=2

(b) -5-3=-8

In above example 5, 5, -3, 8,2 all are integers.

Hence, closure under subtraction is true for integers.

(2) Commutative Property

Integers cannot be subtracted in any order.

Condition for commutative property:

x – y ≠ y – x

Here, x and y are integers.

Example

(a)

5-7 ≠ 7-5

-2 ≠ 2

(b)

6- (-9) ≠ -9-6

6+9 ≠ -9- 6

15 ≠ -15

In both examples, we are not getting same result after subtraction in both sides.

Here, subtraction is not commutative for integers.

(3) Associative Property-

Integers cannot be grouped differently in subtraction.

Condition for associative property-

x – (y – x) = (y – x) – z

Here, x, y and z are integers.

Example

(a) 5- (6-8) can be grouped as (5-6) -8

Now, check the subtraction

5- (6-8) =7 and

(5-6)- 8=-9

Results of both sides are not same.

(b) -3- (-2-5) = [-3 – (-2)]-5

-3+2+5=-3+2-5

4 ≠ – 6

Results of both sides are not same.

Hence, subtraction is not associative for integers.

(4) Identity property-

Subtractive identity is an integer when we subtract it to any integer it gives us same number.

Condition for subtractive identity-

Subtractive identity of an integer states that

x  – identity = x = identity –  x

Example

(a) 7- 0 =7 ≠ 0 -7

(b) -6 – 0=-6 = 0 – (-6)

– 6 = -6 ≠ 6

In both examples when we subtract 0 we do not get same number.

Hence, zero is not subtractive identity for integers.

(5)Distributive property

Integers under subtraction do not have distributive property.

Because, distributive property can only exists when multiplication and addition both involve in an expression. This property is also known as ‘distributive property of multiplication over addition`. It tells us that we distribute the multiplication over all the terms inside the parenthesis or brackets by multiply terms with terms of brackets.

Condition for distributive property:

a ×  (b + c) = ab + ac

a is distribute to b and c by multiplying a inside the terms of brackets.

a – (b + c) will be RHS condition for distributive property of division. We do not have multiplication in this condition. Therefore, it is not possible to distribute division over all the terms inside the parenthesis or brackets.

Hence, distributive property cannot be executed in subtraction.

Helping Topics

Integers