define Properties of Integers with suitable examples( Closure,Commutative,Associative and Distributive
property)
Answers
Answer:
Commutative Property x + y = y+ x x × y = y × x x – y ≠ y – x x ÷ y ≠ y ÷ x
Associative Property x + (y + z) = (x + y) +z x × (y × z) = (x × y) × z (x – y) – z ≠ x – (y – z) (x ÷ y) ÷ z ≠ x ÷ (y ÷ z)
Identity Property x + 0 = x =0 + x x × 1 = x = 1 × x x – 0 = x ≠ 0 – x x ÷ 1 = x ≠ 1 ÷ x
Closure Property x + y ∈ Z x × y ∈ Z x – y ∈ Z x ÷ y ∉ Z
Distributive Property x × (y + z) = x × y + x × z
x × (y − z) = x × y − x × z
The explanation of each of the integer
Step-by-step explanation:
Among the various properties of integers, closure property under addition and subtraction states that the sum or difference of any two integers will always be an integer i.e. if x and y are any two integers, x + y and x − y will also be an integer.
Example 1: 3 – 4 = 3 + (−4) = −1;
(–5) + 8 = 3,
The results are integers.
Closure property under multiplication states that the product of any two integers will be an integer i.e. if x and y are any two integers, xy will also be an integer.
Example 2: 6 × 9 = 54 ; (–5) × (3) = −15, which are integers.
Division of integers doesn’t follow the closure property, i.e. the quotient of any two integers x and y, may or may not be an integer.
The commutative property of addition and multiplication states that the order of terms doesn’t matter, the result will be the same. Whether it is addition or multiplication, swapping of terms will not change the sum or product. Suppose, x and y are any two integers, then
⇒ x + y = y + x
⇒ x × y = y × x
Example 4: 4 + (−6) = −2 = (−6) + 4;
10 × (−3) = −30 = (−3) × 10
But, subtraction (x − y ≠ y − x) and division (x ÷ y ≠ y ÷ x) are not commutative for integers and whole numbers.
Example 5: 4 − (−6) = 10 ; (−6) – 4 = −10
⇒ 4 − (−6) ≠ (−6) – 4
Ex: 10 ÷ 2 = 5 ; 2 ÷ 10 = 1/5
⇒ 10 ÷ 2 ≠ 2 ÷ 10
The associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be the same. One can group numbers in any way but the answer will remain the same. Parenthesis can be done, irrespective of the order of terms. Let x, y and z be any three integers, then
⇒ x + (y + z) = (x + y) +z
⇒ x × (y × z) = (x × y) × z
Example 6: 1 + (2 + (-3)) = 0 = (1 + 2) + (−3);
1 × (2 × (−3)) =−6 = (1 × 2) × (−3)
Subtraction of integers is not associative in nature i.e. x − (y − z) ≠ (x − y) − z.
Example 7: 1 − (2 − (−3)) = −4; (1 – 2) – (−3) = −2
1 – (2 – (−3)) ≠ (1 − 2) − (−3)
The associative property of addition and multiplication states that the way of grouping of numbers doesn’t matter; the result will be the same. One can group numbers in any way but the answer will remain the same. Parenthesis can be done, irrespective of the order of terms. Let x, y and z be any three integers, then
⇒ x + (y + z) = (x + y) +z
⇒ x × (y × z) = (x × y) × z
Example 6: 1 + (2 + (-3)) = 0 = (1 + 2) + (−3);
1 × (2 × (−3)) =−6 = (1 × 2) × (−3)
Subtraction of integers is not associative in nature i.e. x − (y − z) ≠ (x − y) − z.
Example 7: 1 − (2 − (−3)) = −4; (1 – 2) – (−3) = −2
1 – (2 – (−3)) ≠ (1 − 2) − (−3)