# Closure property

Integers are closed under addition, i.e. for any two integers, a and b, a+b is an integer.

Example: 3+4=7, 3 and 4 are integers and when we add them the answer we get is 7 which is also an integer, hence the property.

## Closure property under subtraction

Integers are closed under subtraction, i.e. for any two integers, a and b,
a-b is an integer.
Example: -21 – (-9) = -12, -21 and -9 are integers and when we subtract them the answer we get is -12 which is also an integer, hence the property.

## Closure property under multiplication

Integers are closed under multiplication, i.e. for any two integers,
a and b, ab is an integer.
Example: 5x 6 =30, 5 and 6 are integers and when we multiplied them the answer we get is 30 which is also an integer, hence the property.

## Closure property under division

Integers are NOT closed under division, i.e. for any two integers, a and b, a/b may not be integer.

## Commutative property

Addition is commutative for integers.  For any two integers, a and b,
a + b=b + a
Example: 5 + (-6) = 5 – 6 = 1

(-6)+5 = -6 + 5= -1
∴ 5 + (-6) = (-6) + 5

## Commutative property under subtraction

Subtraction is NOT commutative for integers.  For any two integers, a and b, a – b ≠ b – a

Example: 8- (-6) = 8 + 6 = 14
(-6) – 8 = - 6 – 8 = -14
∴ 8 - (-6) ≠ -6 – 8

## Commutative property under multiplication

Multiplication is commutative for integers.  For any two integers, a and b, ab=ba

Example: 9 × (-6) = -(9×6) = -54
(-6) × 9= - (6×9) = -54
∴ 9 × (-6) = (-6) × 9

## Commutative property under division

Division is NOT commutative for integers.  For any two integers, a and b, a/b ≠ b/a

Example: 3/6=1/2
6/3 = 2
∴ 3/6 ≠ 6/3

## Associative property

Addition is associative for integers.  For any three integers, a, b and c, a+(b+c)=(a+b)+c

Example: 5 + (-6 + 4) = 5 – 2 = 3
(5 - 6) + 4 = -1 + 4 = 3
∴ 5 + (-6+4) = (5 - 6) + 4

## Associative property under subtraction

Subtraction is associative for integers.  For any three integers, a, b and c
a-(b-c) ≠ (a-b)-c

Example:5 - (6-4)=5-2=3;
(5-6)-4=-1-4=-5
∴ 5 - (6-4) ≠ (5-6)-4

## Associative property under multiplication

Multiplication is associative for integers.  For any three integers, a, b and c,
(a×b)×c=a×(b×c)

Example:[(-3)×(-2))×4]=(6×4)=24
[(-3)×(-2×4) ]=(-3×-8)=24
∴ [(-3)×(-2))×4]=[(-3)×(-2×4)]

## Associative property under division

Division is NOT associative for integers.

## Distributive property of multiplication over addition

For any three integers, a, b and c, a×(b+c) = a×b+a×c
Example: -2 (4 + 3) =  -2 (7)  = -14
-2(4+3)=(-2×4)+(-2×3)
=(-8)+(-6)
=-14

## Distributive property of multiplication over subtraction

For any three integers, a, b and c, a×(b-c)= a×b-a×c
Example: -2 (4- 3) =  -2 (1)  = -2
-2(4-3)=(-2×4)-(-2×3)
=(-8)-(-6)
=-2

The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers.

Integer 0 is the identity under addition.  That is, for an integer a, a+0=0+a=a
Example: 4+0=0+4=4

## Identity under multiplication

The integer 1 is the identity under multiplication.  That is, for an integer a, 1×a=a×1=a
Example: (-4)×1=1×(-4)=-4

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