verify all the properties of integers under four binary operations
Answers
HELLO FRIEND!!!!!
- CLOSURE PROPERTY
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 ÷
Hence, set of integer is closed under +,−× but not closed under ÷.
- COMMUTATIVE PROPERTY
COMMUTATIVE PROPERTY OF INTEGERS - DEFINITION
What about subtraction of numbers ?
Take a and b as two integers and subtract them i.e. a−b.
Now, subtract a from b i.e. b−a.
Are they same ?
No, they are not equal.
So, commutative property does not hold for subtraction.
Similarly, it does not hold for division.
- 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.
- 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.