Define closure law and commutative law for integers with example
Answers
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Closure Property under Addition of Integers
If we add any two integers, the result obtained on adding the two integers, is always an integer. So we can say, that integers are closed under addition.
Let us say ‘a’ and ‘b’ are two integers, either positive or negative. When we add the two integers, their result would always be an integer, i.e (a + b) would always be an integer.
Example –
State whether (– 11) + 2 is closed under addition
Solution
– 11 + 2
– 9
Since both -11 and 2 are integers, and their sum, i.e (-9) is also an integer, we can say that integers are closed under addition.
Closure Property under Subtraction of Integers
If we subtract any two integers the result is always an integer, so we can say that integers are closed under subtraction.
Let us say ‘a’ and ‘b’ are two integers either positive or negative, their result should always be an integer, i.e (a + b) would always be an integer.
Example –
State whether (24 – 12) is closed under subtraction
Solution –
24 – 12
12
Since both 24 and -12 are integers, and their difference, i.e (12) is also an integer, we can say that integers are closed under subtraction.
Closure Property under Multiplication of Integers:
If we multiply any two integers the result is always an integer, so we can say that integers are closed under multiplication.
Let us say ‘a’ and ‘b’ are two integers either positive or negative, and if multiply it, their result should always be an integer, i.e [(-a) x b] and [a x (–b)] would always be an integer.
Example –
Show that (-30) x 11 closed under multiplication
Solution –
– 30 x 11
-330
Since both -30 and 11 are integers, and their product, i.e (-330) is also an integer, we can say that integers are closed under multiplication.
Closure Property under Division of Integers:
If we divide any two integers the result is not necessarily an integer, so we can say that integers are not closed under division.
Let us say ‘a’ and ‘b’ are two integers, and if we divide them, their result ( a ÷ b ) is not necessarily an integer.
Example –
State whether (14) ÷ 5 is closed under division.
Solution –
(14) ÷ 5
8
Since both 14 and 5 are integers, but (14) ÷ 5 = 2.8 which is not an integer. Hence, we can say that integers are not closed under division.
Commutative Property Of Integers

Commutative Property under Addition of Integers:
If we add two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we add (2+3) = (3+2) = 5. So whole numbers are commutative under addition. Similarly if we apply this to integers, (-5+3) = (3+(-5))= -2, it also hold for all integers. So we can say that commutative property holds under addition for all integers.
In generalise form for any two integers ‘a’ and ‘b’
a + b = b + a
Example –
Show that -32 and 23 follow commutative property under addition.
Solution –
L.H.S = -32 + 23 = – 9
R.H.S = 23 + (-32) = 23 – 32 = – 9
So, L.H.S = R.H.S, i.e a + b = b + a
This means the two integers follow commutative property under addition.
Commutative Property under Subtraction of Integers:
On contradictory, commutative property will not hold for subtraction of whole number say (5 – 6) is not equal to (6 – 5). Let us consider for integers (4) and (-1), the difference of two numbers are not always same.
{4 – (-1) = 4 + 1= 5} and {(-1) – 4 = – 1 – 4 = -5}, so the difference of two integers are 5 and (-5) which are not equal so we can say that commutative property will not hold for subtraction of integers.
In generalise form for any two integers ‘a’ and ‘b’
(a – b) ≠ (b – a)
Example –
Check whether -88 and 22 follow commutative property under subtraction.
Solution –
L.H.S = -88 – 22 = – 110
R.H.S = 22 – (-88) = 22 + 88 = 110
So, L.H.S ≠ R.H.S
This means the two integers do not follow commutative property under subtraction.
Commutative Property under Multiplication of Integers:
If we multiply two whole numbers say ‘a’ and ‘b’ the answer will always same, i.e if we multiply (2×3) = (3×2) = 6. So whole numbers are commutative under multiplication. Similarly if we apply this to integers, (-5×3) = (3x(-5))= -6, it also hold true for all integers. So we can say that commutative property holds under l
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