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**Closure property**

**Closure property under addition:**

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

Ex: 3 + 4 = 7; (â€“ 9) + 7 = â€“ 2.

**Closure property under subtraction:**

Integers are closed under subtraction, i.e. for any two integers a and b, a â€“ b is an integer.

Ex: (â€“ 21) â€“ (â€“ 9) = (â€“ 12); 8 â€“ 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 Ã— 6 = 30; (â€“ 9) Ã— (â€“ 3) = 27.

**Closure property under division:**

Integers are not closed under division, i.e. for any two integers a and b, $\frac{\text{a}}{\text{b}}$may not be an integer.

Ex:(â€“ 2) Ã· (â€“ 4) = $\frac{\text{1}}{\text{2}}$

**Commutative property**

**Commutative property under addition:**

Addition is commutative for integers. For any two integers a and b, a + b = b + a.

Ex: 5 + (â€“ 6) = 5 â€“ 6 = â€“ 1;

(â€“ 6) + 5 = â€“ 6 + 5 = â€“1

âˆ´ 5 + (â€“ 6) = (â€“ 6) + 5.

**Commutative property under subtraction:**

Subtraction is not commutative for integers. For any two integers a and b, a â€“ b â‰ b â€“ a.

Ex: 8 â€“ (â€“ 6) = 8 + 6 = 14;

(â€“ 6) â€“ 8 = â€“ 6 â€“ 8 = â€“ 14

âˆ´ 8 â€“ (â€“ 6) â‰ â€“ 6 â€“ 8.

**Commutative property under multiplication:**

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

Ex: 9 Ã— (â€“ 6) = â€“ (9 Ã— 6) = â€“ 54;

(â€“ 6) Ã— 9 = â€“ (6 Ã— 9) = â€“ 54

âˆ´ 9 Ã— (â€“ 6) = (â€“ 6) Ã— 9.

**Commutative property under division:**

Division is not commutative for integers. For any two integers a and b, a Ã· b â‰ b Ã· a.

Ex: (â€“ 14) Ã· 2 = â€“ 7

2 Ã· (â€“14) = â€“ $\frac{\text{1}}{\text{7}}$

(â€“ 14) Ã· 2 â‰ 2 Ã· (â€“14).

**Associative property**

**Associative property under addition:**

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

Ex: 5 + (â€“ 6 + 4) = 5 + (â€“ 2) = 3;

(5 â€“ 6) + 4 = (â€“ 1) + 4 = 3

âˆ´ 5 + (â€“ 6 + 4) = (5 â€“ 6) + 4.

**Associative property under subtraction:**

Subtraction is associative for integers. For any three integers a, b and c, a â€“ (b â€“ c) â‰ (a â€“ b) â€“ c

Ex: 5 â€“ (6 â€“ 4) = 5 â€“ 2 = 3;

(5 â€“ 6) â€“ 4 = â€“ 1 â€“ 4 = â€“ 5

âˆ´ 5 â€“ (6 â€“ 4) â‰ (5 â€“ 6) â€“ 4.

**Associative property under multiplication:**

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

Ex: [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = (6 Ã— 4) = 24

(â€“ 3) Ã— [(â€“ 2) Ã— 4] = (â€“ 3) Ã— (â€“ 8) = 24

âˆ´ [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = [(â€“ 3) Ã— (â€“ 2) Ã— 4].

**Associative property under division:**

Division is not associative for integers.

**Distributive property**

**Distributive property of multiplication over addition:**

For any three integers a, b and c, a Ã— (b + c) = (a Ã— b) + (a Ã— c).

Ex: â€“ 2 (4 + 3) = â€“2 (7) = â€“14

= (â€“ 2 Ã— 4) + (â€“ 2 Ã— 3)

= (â€“ 8) + (â€“ 6)

= â€“ 14.

**Distributive property of multiplication over subtraction:**

For any three integers, a, b and c, a Ã— (b - c) = (a Ã— b) â€“ (a Ã— c).

Ex: â€“ 2 (4 â€“ 3) = â€“ 2 (1) = â€“ 2

= (â€“2 Ã— 4) â€“ (â€“ 2 Ã— 3)

= (â€“ 8) â€“ (â€“ 6)

= â€“ 2.

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

**Identity under addition:**

Integer 0 is the identity under addition. That is, for any integer a, a + 0 = 0 + a = a*.*

Ex: 4 + 0 = 0 + 4 = 4.

**Identity under multiplication:**

The integer 1 is the identity under multiplication. That is, for any integer a, 1 Ã— a = a Ã— 1 = a.

Ex: (â€“ 4) Ã— 1 = 1 Ã— (â€“ 4) = â€“ 4.

When an integer is multiplied by â€“1, the result is the integer with sign changed i.e. the additive identity of the integer.

For any integer a, a Ã— â€“1 = â€“1 Ã— a = â€“a.

**Closure property**

**Closure property under addition:**

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

Ex: 3 + 4 = 7; (â€“ 9) + 7 = â€“ 2.

**Closure property under subtraction:**

Integers are closed under subtraction, i.e. for any two integers a and b, a â€“ b is an integer.

Ex: (â€“ 21) â€“ (â€“ 9) = (â€“ 12); 8 â€“ 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 Ã— 6 = 30; (â€“ 9) Ã— (â€“ 3) = 27.

**Closure property under division:**

Integers are not closed under division, i.e. for any two integers a and b, $\frac{\text{a}}{\text{b}}$may not be an integer.

Ex:(â€“ 2) Ã· (â€“ 4) = $\frac{\text{1}}{\text{2}}$

**Commutative property**

**Commutative property under addition:**

Addition is commutative for integers. For any two integers a and b, a + b = b + a.

Ex: 5 + (â€“ 6) = 5 â€“ 6 = â€“ 1;

(â€“ 6) + 5 = â€“ 6 + 5 = â€“1

âˆ´ 5 + (â€“ 6) = (â€“ 6) + 5.

**Commutative property under subtraction:**

Subtraction is not commutative for integers. For any two integers a and b, a â€“ b â‰ b â€“ a.

Ex: 8 â€“ (â€“ 6) = 8 + 6 = 14;

(â€“ 6) â€“ 8 = â€“ 6 â€“ 8 = â€“ 14

âˆ´ 8 â€“ (â€“ 6) â‰ â€“ 6 â€“ 8.

**Commutative property under multiplication:**

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

Ex: 9 Ã— (â€“ 6) = â€“ (9 Ã— 6) = â€“ 54;

(â€“ 6) Ã— 9 = â€“ (6 Ã— 9) = â€“ 54

âˆ´ 9 Ã— (â€“ 6) = (â€“ 6) Ã— 9.

**Commutative property under division:**

Division is not commutative for integers. For any two integers a and b, a Ã· b â‰ b Ã· a.

Ex: (â€“ 14) Ã· 2 = â€“ 7

2 Ã· (â€“14) = â€“ $\frac{\text{1}}{\text{7}}$

(â€“ 14) Ã· 2 â‰ 2 Ã· (â€“14).

**Associative property**

**Associative property under addition:**

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

Ex: 5 + (â€“ 6 + 4) = 5 + (â€“ 2) = 3;

(5 â€“ 6) + 4 = (â€“ 1) + 4 = 3

âˆ´ 5 + (â€“ 6 + 4) = (5 â€“ 6) + 4.

**Associative property under subtraction:**

Subtraction is associative for integers. For any three integers a, b and c, a â€“ (b â€“ c) â‰ (a â€“ b) â€“ c

Ex: 5 â€“ (6 â€“ 4) = 5 â€“ 2 = 3;

(5 â€“ 6) â€“ 4 = â€“ 1 â€“ 4 = â€“ 5

âˆ´ 5 â€“ (6 â€“ 4) â‰ (5 â€“ 6) â€“ 4.

**Associative property under multiplication:**

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

Ex: [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = (6 Ã— 4) = 24

(â€“ 3) Ã— [(â€“ 2) Ã— 4] = (â€“ 3) Ã— (â€“ 8) = 24

âˆ´ [(â€“ 3) Ã— (â€“ 2)] Ã— 4 = [(â€“ 3) Ã— (â€“ 2) Ã— 4].

**Associative property under division:**

Division is not associative for integers.

**Distributive property**

**Distributive property of multiplication over addition:**

For any three integers a, b and c, a Ã— (b + c) = (a Ã— b) + (a Ã— c).

Ex: â€“ 2 (4 + 3) = â€“2 (7) = â€“14

= (â€“ 2 Ã— 4) + (â€“ 2 Ã— 3)

= (â€“ 8) + (â€“ 6)

= â€“ 14.

**Distributive property of multiplication over subtraction:**

For any three integers, a, b and c, a Ã— (b - c) = (a Ã— b) â€“ (a Ã— c).

Ex: â€“ 2 (4 â€“ 3) = â€“ 2 (1) = â€“ 2

= (â€“2 Ã— 4) â€“ (â€“ 2 Ã— 3)

= (â€“ 8) â€“ (â€“ 6)

= â€“ 2.

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

**Identity under addition:**

Integer 0 is the identity under addition. That is, for any integer a, a + 0 = 0 + a = a*.*

Ex: 4 + 0 = 0 + 4 = 4.

**Identity under multiplication:**

The integer 1 is the identity under multiplication. That is, for any integer a, 1 Ã— a = a Ã— 1 = a.

Ex: (â€“ 4) Ã— 1 = 1 Ã— (â€“ 4) = â€“ 4.

When an integer is multiplied by â€“1, the result is the integer with sign changed i.e. the additive identity of the integer.

For any integer a, a Ã— â€“1 = â€“1 Ã— a = â€“a.