Home
Class 12
MATHS
Let S={a+sqrt(2)\ b\ : a ,\ b in Z}dot ...

Let `S={a+sqrt(2)\ b\ : a ,\ b in Z}dot` Then, prove that an operation * on `S` defined by `(a_1+sqrt(2)b_1)*(a_2+sqrt(2)b_2)=(a_1+a_2)+sqrt(2)(b_1+b_2)` for all `a_1,\ a_2,\ b_1,\ b_2 in Z` is a binary operation on `Sdot`

Text Solution

Verified by Experts

`(a_1+sqrt(2)b_1)*(a_2+sqrt(2)b_2)`
`=>(a_1+a_2)+sqrt2(b_1+b_2)`
here,`(a_1+a_2)inZ`
and also `sqrt2(b_1+b_2)inZ`
so,`(a_1+a_2)+sqrt2(b_1+b_2)in S`
Hence * is a binary operation.
Promotional Banner

Topper's Solved these Questions

  • ARITHMETIC PROGRESSION

    RD SHARMA|Exercise EXAMPLE|5 Videos

  • BINOMIAL DISTRIBUTION

    RD SHARMA|Exercise Solved Examples And Exercises|141 Videos

Similar Questions

Explore conceptually related problems

Let S={a+sqrt(2)b:a,b in Z}. Then prove that an operation * on S defined by (a_(1)+sqrt(2)b_(1))*(a_(2)+sqrt(2)b_(2))=(a_(1)+b_(2))+sqrt(2)(b_(1)+b_(2)) for all b_(1),a_(2)in Z is binary operation ofn S.

If the points (a_1, b_1),\ \ (a_2, b_2) and (a_1+a_2,\ b_1+b_2) are collinear, show that a_1b_2=a_2b_1 .

If (a_2 a_3)/(a_1 a_4) = (a_2 + a_3)/(a_1 + a_4) = 3 ((a_2 - a_3)/(a_1 - a_4)) then a_1, a_2, a_3, a_4 are in

Prove by vector method that (a_1b_1+a_2b_2+a_3b_3)^2lt+(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)

If the equation of the locus of a point equidistant from the points (a_1,b_1) and (a_2,b_2) is (a_1-a_2)x+(b_2+b_2)y+c=0 , then the value of C is

If (b_2-b_1)(b_3-b-1)+(a_2-a_1)(a_3-a_1)=0 , then prove that the circumcenter of the triangle having vertices (a_1,b_1),(a_2,b_2) and (a_3,b_3) is ((a_2+a_3)/(2),(b_2+b_3)/(2)) .

In direct proportion a_1/b_1 = a_2/b_2

RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
  1. Let * be a binary operation o Q defined by a*b= (ab)/4 for all a,bin Q...

    Text Solution

    |

  2. If the binary operation o. is defined on the set Q^+ of all positive r...

    Text Solution

    |

  3. Let S={a+sqrt(2)\ b\ : a ,\ b in Z}dot Then, prove that an operati...

    Text Solution

    |

  4. Let S={1,\ 2,\ 3,\ 4} and * be an operation on S defined by a*b=r , wh...

    Text Solution

    |

  5. Let S=(0,1,2,3,4,) and * be an operation on S defined by a*b=r , where...

    Text Solution

    |

  6. Show that the operation vv and ^^ on R defined as avvb= Maximum of ...

    Text Solution

    |

  7. On the set Q of all rational numbers an operation * is defined by a*b ...

    Text Solution

    |

  8. On the set W of all non-negative integers * is defined by a*b=a^b ....

    Text Solution

    |

  9. On the set C of all complex numbers an operation 'o' is defined by ...

    Text Solution

    |

  10. Let M be the set of all singular matrices of the form [xxxx] , wher...

    Text Solution

    |

  11. Determine whether * on N defined by a*b=a^b for all a ,\ b in N de...

    Text Solution

    |

  12. Determine whether O on Z defined by a\ O\ b=a^b for all a ,\ b in ...

    Text Solution

    |

  13. Determine whether * on N defined by a*b=a+b-2 for all a ,\ b in N ...

    Text Solution

    |

  14. Determine whether 'xx6' on S={1,\ 2,\ 3,\ 4,\ 5} defined by axx6b= Rem...

    Text Solution

    |

  15. Determine whether '+6' on S={0,\ 1,\ 2,\ 3,\ 4,\ 5} defined by a+6b={a...

    Text Solution

    |

  16. 'o' on N defined by a o b= a b + b a for all a, b∈N define...

    Text Solution

    |

  17. '*' on Q defined by a*b=(a-1)/(b+1) for all a ,\ b in Q define a bina...

    Text Solution

    |

  18. Determine whether or not the definition of * On Z^+ , defined * by ...

    Text Solution

    |

  19. Determine whether or not the definition of * On Z^+ , defined * by ...

    Text Solution

    |

  20. Determine whether or not the definition of * On R , define by a*b=a...

    Text Solution

    |