Home
Class 12
MATHS
Define a binary operation ** on the set ...

Define a binary operation `**` on the set `{0, 1, 2, 3, 4, 5}` as
`a**b={(a+b , if a+b<6;),(a+b-6, if a+b ge 6):}`.
Show that zero is the identity for this operation and each element `a !=0` of the set is invertible with `6-a` being the inverse of `a`

Text Solution

Verified by Experts

Given `X={0,1,2,3,4,5}`
`a**b={a+b,if a+b<6;` `a+b-6,if a+b geq6`
To check if zero is the identity , we see that `a**0=a+0=a`
for all `a in x ` and also `0**a=0+a=a` for `a in x`
Given `a in x ,a+0<6 ` and also `0+a<6`
0 is the identity element for the given operation .
Now
The element `a in x` is invertible if there exists `b in x` such that `a xx b=e=b**a`
In this case , e=0 to `a**b=0=b**a`
`implies a**b ={a+b=0=b+a,if a+b<6;a+b-6=0=b+a-6 if a+b geq 6}`
`i.e. a=-b or b=6-a`
but since , `ab in x ={0,1,2,3,4,5}`
Hence `b=6-a` is the inverse of `a` i.e.`6-a`
`AA a in {1,2,3,4,5}`
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise SOLVED EXAMPLES|50 Videos

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise EXERCISE 1.4|13 Videos

  • PROBABILITY

    NCERT|Exercise EXERCISE 13.2|18 Videos

  • SETS

    NCERT|Exercise EXERCISE 1.3|1 Videos

Similar Questions

Explore conceptually related problems

Define a binary operation ** on the set A={0,1,2,3,4,5} as a**b=(a+b) \ (mod 6) . Show that zero is the identity for this operation and each element a of the set is invertible with 6-a being the inverse of a . OR A binary operation ** on the set {0,1,2,3,4,5} is defined as a**b={[a+b if a+b = 6]} Show that zero is the identity for this operation and each element a of the set is invertible with 6-a , being the inverse of a .

A binary operation * on the set {0,1,2,3,4,5} is defined as: a*b={a+b a+b-6" if "a+b<6" if" a+bgeq6 Show that zero is the identity for this operation and each element a of the set is invertible with 6a, being the inverse of a.

Define a binary operation * on the set A={0,1,2,3,4,5} as a*b=a+b( mod 6). Show that zero is the identity for this operation and each element a of the set is invertible with 6-a being the inverse of a.

Define a binary operation * on the set A={1,2,3,4} as a^(*)b=ab(mod5). Show that 1 is the identity for * and all elements of the set A are invertible with 2^(-1)=3 and 4^(-1)=4

Define a binary operation * on the set A={1,2,3,4} as a*b=ab(mod5) show that 1 is the identity for * and all elements of the set A are invertible with 2^(-1)=3 and 4^(-1)=4

Consider the binary operation* on the set {1, 2, 3, 4, 5} defined by a * b=min. {a, b}. Write the operation table of the operation *.

Define a binary operation * on the set A={0,1,2,3,4,5} given by a*b=ab(mod 6).Show that 1 is the identity for *.1 and 5 are the only invertible elements with 1^(-1)=1 and 5^(-1)=5

If the binary operation ** on the set Z of integers is defined by a**b=a+b-5 , then write the identity element for the operation '**' in Z.

If the binary operation * on the set Z of integers is defined by a*b=a+b-5 then write the identity element for the operation * in Z .