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
Answers
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:
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: