## HOW TO UNDERSTAND BINARY OPERATIONS IN RELATIONS AND FUNCTIONS

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Then how will you solve this problem or such types of problems? so with the of help Binary operations we can solve such problems,

##

If a person leaves for his office at 9 am daily ,which is 5 KM from his home , and comes back home at 6 pm , then its distance from home to office and back office to home is same 5 KM , then this Property is called commutative Property .

For example

(1) If Shayam slaps Ram twice , and in return Ram also slaps Shyam then it is called commutative.

(2) if we add 4 to 5 we shall get 9 and in other case if we add 5 to 4 then we also get 9. implies 4 + 5 = 5 + 4 = 9

In Mathematics it written as

but if we subtract 4 from 5 we shall get 1 but if we subtract 5 from 4 we shall get -1, then this relation is

##

If we have three numbers 4, 5, and 6 , and we have to add these three numbers in two ways that 1st we add 4 and 5 and then 6 will be added to the result obtained in last step. and in second ways we shall 5 and 6 then resultant will be added to 4 . And in both the cases result comes out same then this will be called Associated property.

Today we shall discuss Binary operation and how to understand binary operation with the help of some Examples. Because with the help of Binary operations we can crack many quizzes like .

**3 4 15**

##
** Commutative ****Property**

For example

(1) If Shayam slaps Ram twice , and in return Ram also slaps Shyam then it is called commutative.

(2) if we add 4 to 5 we shall get 9 and in other case if we add 5 to 4 then we also get 9. implies 4 + 5 = 5 + 4 = 9

In Mathematics it written as

**a ∗ b = b∗ a for all values of a ,b .**

but if we subtract 4 from 5 we shall get 1 but if we subtract 5 from 4 we shall get -1, then this relation is

**not**called commutative ,because answer are

**not same**in both the cases.

##
**Associative ****Property**

( 4 + 5 ) + 6 = 4 + ( 5 + 6 )

9 + 6 = 4 + 11 =15

But if we subtract these numbers in place of sum ,then these numbers do not satisfy the property of Associativity .

( 4 - 5 ) - 6 ≠ 4 - ( 5 - 6 )

(-1) - 6 ≠ 4 - (-1)

-1 - 6 ≠ 4 + 1

-7 ≠ 5

This concept can be better understand with the help of this video

This concept can be better understand with the help of this video

## Binary Operations

A binary operation ∗ on a set A is a function ∗ : A × A → A. We denote ∗ (a, b) by a ∗ b## Question

Let ∗ be binary operation on the set Q of rational numbers defined below

(1) a ∗ b = a – b

(2) a ∗ b = ab + 1

(3) a ∗ b = a + ab

(4) a * b = | a - b |

Determine whether ∗ is binary, commutative or associative.

####
**Solution**

(1) a ∗ b = a – b###
** Commutative ****Property**

a ∗ b
= b∗ a

Now

a ∗ b
= a – b -----------------(1)

b*a = b -
a = - (a-b) -----------(2)

Therefore
a ∗ b ≠ b∗ a

Therefore
* is not a commutative operation under Q

###
**Associative ****Property**

(a ∗ b) *c
= a∗ (b*c)

(a ∗ b )*c = (a – b)*c
Replaced a*b with a - b as given

Now assume
"a-b" as 1st number(blue) and
"c" as 2nd number(red) Again
using eq (1)

(a ∗ b )*c =(a – b) * c

**=**(a - b ) - c

= a - b - c ------------(3)

From (1) and (2)

a * (b * c) = a*(b - c)

Now
assume "a" as 1st number(blue) and "b - c" as 2nd number(red)

a * (b * c)
= (a) - (b - c )

= a - b + c ------------(4)

From (1) and (2)

(a ∗ b)
*c ≠ a ∗ (b * c)

Therefore * is not a associative operation under Q

##
**(2)
a ****∗**** b = ab + 1 (Product of two
numbers and plus one)**

###
**For commutative Property**

a ∗ b = b∗ a

Now

a ∗ b =

**ab + 1**------------------(5)
b ∗ a = ba + 1 = ab +1 --------(6)

Therefore a ∗ b = b∗a

Therefore * is a commutative operation under Q

###
**For Associative ****Property**

(a ∗ b) *c = a∗ (b*c)

(a
∗ b )*c =(ab + 1) * c
Replaced a*b with ab+1 as given

Now assume " ab+1 " as 1st number(blue) and "c"
as 2nd number(red)

(a ∗ b )*c =(ab + 1)*c

**(Product of two numbers and plus one)****=**(ab + 1)(c) + 1

=
(ab + 1)c +1

= abc + c + 1 -------(7)

a * (b*c) = a*(bc + 1)

**(Product of two numbers and plus one)**
Now assume "a" as 1st number(blue) and "bc+1" as 2nd number(red)

a * (b*c) = (a)(bc + 1) + 1

=abc + a + 1
------------------------(8)

From (7) and (8)

(a ∗ b) *c ≠ a∗ (b*c)

Therefore * is not a associative operation under Q

##
**(3)
****a
****∗**** b =****a + ab ** **(Product of two
numbers and plus 1st num )**

### so if we put all the values of aand b according to table then we can solve problem discussed in begining of the post.

Binary Operations |

###
For **commutative ****Property**

a ∗ b = b ∗ a

Now

a ∗ b =

**a + ab**-----------------(9)
b ∗ a = b + ba = b + ab
-------(10)

Therefore a ∗ b ≠ b ∗ a

Therefore * is a not commutative operation under Q

###
**For Associativeness**

(a ∗ b) *c = a∗ (b*c)

(a
∗ b )*c =(a+ab)*c
Replaced a*b with a+ab as given

Now assume "a + ab" as 1st number(blue) and "c"
as 2nd number(red)

(a ∗ b )*c =(a + ab) * c (Product of two numbers and plus 1st number , here 1st num is a + ab and 2nd num is c)

**=**(a + ab ) + ( a + ab )c

=
a + ab + ( a + ab )c

= a + ab + ac + abc
------------------------(11)

a * (b*c) = a*( b + bc)

**(**Product of two numbers and plus 1st number**)**
Now assume "a" as 1st number(blue) and "b + bc" as 2nd number(red)

a * (b * c) = (a) + a( b + bc)

=a + a( b + bc)

=a + ab + abc
------------------------(12)

From (11) and (12)

(a ∗ b) *c ≠ a∗ (b*c)

Therefore * is not a associative operation under Q

(4)

For

**commutative****Property**

a ∗ b = b ∗ a

Now

a ∗ b = |a - b | ---------------------(13)

b ∗ a = |b - a |= |a - b | ---------------(14)

Therefore a ∗ b = b ∗ a

Therefore * is a commutative operation under Q

###

###
**For Associativeness**

( a ∗ b ) * c = a ∗ (b * c)

(a ∗ b )*c = (| a - b | ) * c Replaced a*b with | a - b | as given

Now assume "| a - b |" as 1st number and "c" as 2nd number

(a ∗ b )*c =(| a - b | ) * c ( modulus of difference of two numbers , here 1st num is | a - b | and 2nd num is c)

**= |**(| a - b |) - c |

=

**|**| a - b | - c | ------------------------(15)
a * (b*c) = a* ( |b-c |)

**(**Modulus of difference of two numbers**)**
Now assume "a" as 1st number(blue) and "|b-c |" as 2nd number(red)

a * (b * c) = | (a) - |b-c | |

=

= | a - |b-c | | -------- (16)

From (15) and (16)

(a ∗ b) *c ≠ a∗ (b*c)

Therefore * is not a associative operation under Q

## Conclusion

Thanks for devoting your precious time to read this post. I hope this post on How to understand Binary Operations , commutative , Associative has helped you more , If you find this post little bit of your concern then, then follow me on my blog and read my other posts . We shall meet in next post, till then Bye.

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