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Ex: 3 + 4 = 7; (– 9) + 7 = – 2.

Closure property under subtraction:
Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer.
Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5.

Closure property under multiplication:
Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer.
Ex: 5 × 6 = 30; (– 9) × (– 3) = 27.

Closure property under division:
Integers are not closed under division, i.e. for any two integers a and b, a/b may not be an
integer.
Ex:(– 2) ÷ (– 4) = 1/2

Commutative property
Commutative property under addition:
Addition is commutative for integers. For any two integers a and b, a + b = b + a.

Ex: 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.
Ex: 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.
Ex: 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.
Ex: (– 14) ÷ 2 = – 7

2 ÷ (–14) = – 1717
(– 14) ÷ 2 ≠ 2 ÷ (–14).

Associative property

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