Question

we now study the properties satisfied by addition and subtraction. (a) integers are closed for addition and subtraction both. that is,a+b and a-b are again integers,where a and b are any integers​

Answer 1

Answer:

Step-by-step explanation:

y of an integer is a property which states that when we add any integer a with 0 then the resultant is the number itself.

Example: 6+0=6  

MULTIPLICATIVE IDENTITY - DEFINITION

Multiplicative Identity of integer is a property which states that the product of any integer and digit 1 is the number itself.

Example: let 5 be an integer then 5×1=1×5=5

CLOSURE PROPERTY OF INTEGERS - DEFINITION

Closure property says that if for any two integers a and b, a∗b is also an integer then the set of integers is closed under ∗

where ∗ represents +,−,× or ÷

For example:- Take 4 and 8

Now, 4+8=12 is an integer

4×8=32 is an integer

4−8=−4 is an integer

But 4÷8=  

8

4

​  

=  

2

1

​  

 is not an integer

Hence, set of integer is closed under +,−× but not closed under ÷.

ASSOCIATIVE PROPERTY OF INTEGERS - DEFINITION

Associative property states that, for any three elements(numbers) a,b and c we have

a∗(b∗c)=(a∗b)∗c, where ∗ represents a binary operation.

Let's take ∗ as addition(+)

Then, we have a+(b+c)=(a+b)+c

For eg:- For 2,5 and 11

2+(5+11)=2+16=18 and (2+5)+11=7+11=18

For multiplication

2×(5×11)=2×55=110 and (2×5)×11=10×11=110

Hence, a∗(b∗c)=(a∗b)∗c is true for addition and multiplication.

ASSOCIATIVE PROPERTY OF INTEGERS - EXAMPLE

What about subtraction and division ?

Associative property does not hold for subtraction and division

Let's take an example :

For 4,6 and 12

4÷(6÷12)=4÷  

12

6

​  

=4÷  

2

1

​  

=  

2

1

​  

 

4

​  

=4×2=8 and  

(4÷6)÷12=  

6

4

​  

÷12=  

3

2

​  

÷12=  

12

3

2

​  

 

​  

=  

3×12

2

​  

=  

3×6

1

​  

=  

18

1

​  

 



=8

Therefore, a∗(b∗c)=(a∗b)∗c is not true for division.

Also, 4−(6−12)=4−(−6)=4+6=10 and (4−6)−12=−2−12=−14



=10

Hence, a∗(b∗c)=(a∗b)∗c is not true for subtraction as well.

ADDITIVE IDENTITY FOR INTEGERS - DEFINITION

Additive identity of any integer a is a number b which when added to a, leaves it unchanged.

i.e. b is called as additive identity of any integer a if a+b=a

When we add 0 to any of the integer a, we get

a+0=a=0+a

So, 0 is the additive identity for integers.

For eg:- 2+0=2 and 0+2=2

MULTIPLICATIVE IDENTITY FOR INTEGERS - DEFINITION

Multiplicative identity of any integer a is a number b which when multiplied with a, leaves it unchanged.

i.e. b is called as multiplicative identity of any integer a if a×b=a

Now, when we multiply 1 with any of the integer a, we get

a×1=a=1×a

So, 1 is the multiplicative identity for integers.

For eg:- 3×1=3 and 1×3=3

CLOSURE PROPERTY IN REFERENCE TO INTEGERS - DEFINITION

System of Integers under Addition:

Addition of two Integers always results in an Integer.

Eg:

7+4=11, Result is an Integer.

Therefore, system is closed under addition.

System of Integers under Subtraction:

Subtraction of two Integers always results in an Integer.

Eg:

7−4=3, Result is an Integer, and  

2−4=−2, Result is also an integer.

Therefore, system is closed under subtraction.

System of integers under Multiplication:

Multiplication of two integers always results in an integers.

Eg:

7×4=28, Result is an Integer

Therefore, system is closed under Multiplication.

System of Integers under Division:

Division of two integers does not always results in an integer.

Eg:

8÷4=2, Result is an integer, but

7÷4=  

4

7

​  

, Result is not an integer.

Therefore, system is not closed under division.

Answer 2

Answer:

integers are closed for ______ and ______ both. that is a+b and a-b are again integers ,where a and b are any integers

Step-by-step explanation: