Home
Class 12
MATHS
Let R0 denote the set of all non-zero...

Let `R_0` denote the set of all non-zero real numbers and let `A=R_0xxR_0` . If * is a binary operation on `A` defined by `(a ,\ b)*(c ,\ d)=(a c ,\ b d)` for all `(a ,\ b),\ (c ,\ d) in Adot` Find the invertible element in `A` .

Text Solution

Verified by Experts

Let `({m}, {n})` be the inverse of `(a, b) forall(a, b) in A`. Then,
`(a, b) *(m, n)=(1,1)`
`Rightarrow(a m, b n)=(1,1) `
`Rightarrow a m=1` & `b n=1`
`Rightarrow m=frac{1}{a}` & `n=frac{1}{b}`
Thus, `(frac{1}{a}, frac{1}{b})` is the inverse of `(a, b) forall(a, b) in A`.
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 R_(0) denote the set of all non-zero real numbers and let A=R_(0)xx R_(0) .If * is a binary operation on A defined by (a,b)*(c,d)=(ac,bd) for all (a,b),(c,d)in A* show that * is both commutative and associative on A( ii) Find the identity element in A

Let A=Q xx Q and let * be a binary operation on A defined by (a,b)*(c,d)=(ac,b+ad) for (a,b),(c,d)in A. Then,with respect to * on A* Find the invertible elements of A.

Let A=NxxN , and let * be a binary operation on A defined by (a , b)*(c , d)=(a d+b c , b d) for all (a , b), (c , d) in NxxNdot Show that: * is commutative on Adot (ii) * is associative on Adot

Let A=N uu{0}xx N uu{0} and let * be a binary operation on A defined by (a,b)*(c,d)=(a+c,b+d) for all (a,b),(c,d)in A. Show that * is commutative on A.

Let A=N uu{0}xx N uu{0} and let * be a binary operation on A defined by (a,b)*(c,d)=(a+c,b+d) for all (a,b),(c,d)in A* show that * is associative on A.

Let A = Q xx Q, where Q is the set of all rational numbers, and * be abinary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) in A.Find the identity element in A.

Let A=N xx N, and let * be a binary operation on A defined by (a,b)*(c,d)=(ad+bc,bd) for all (a,b),(c,d)in N xx N* Show that A has no identity element.

Let A=Q xx Q and let * be a binary operation on A defined by (a,b)*(c,d)=(ac,b+ad) for (a,b),(c,d)in A. Then,with respect to * on A* Find the identity element in A

Let A=QxQ and let * be a binary operation on A defined by (a,b)*(c,d)=(ac,b+ad) for (a,b),(c,d)in A. Then,with respect to * on A Find the identity element in A Find the invertible elements of A.

Let A=NxNa n d^(prime)*' be a binaryoperation on A defined by (a , b)*(C , d)=(a c , b d) for all a , b , c , d , in Ndot Show that '*' is commutative and associative binary operation on A.

RD SHARMA-BINARY OPERATIONS-Solved Examples And Exercises
  1. On R-[1] , a binary operation * is defined by a*b=a+b-a b . Prove that...

    Text Solution

    |

  2. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

    Text Solution

    |

  3. Let R0 denote the set of all non-zero real numbers and let A=R0xxR0...

    Text Solution

    |

  4. Let * be the binary operation on N defined by a*b=H C F of a and b ...

    Text Solution

    |

  5. Consider the set S={1,\ -1} of square roots of unity and multiplica...

    Text Solution

    |

  6. Consider the set S={1,\ omega,\ omega^2} of all cube roots of unity...

    Text Solution

    |

  7. Consider the set S={1,\ -1,\ i ,\ -i} of fourth roots of unity. Con...

    Text Solution

    |

  8. Consider the set S={1,\ 2,\ 3,\ 4} . Define a binary operation * on...

    Text Solution

    |

  9. \begin{tabular}{|l|l|l|l|l|l|} \hline 1 & 1 & 2 & 3 & 4 & 5 \\ \hline ...

    Text Solution

    |

  10. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by t...

    Text Solution

    |

  11. Define a binary operation * on the set A={0,\ 1,\ 2,\ 3,\ 4,\ 5} as...

    Text Solution

    |

  12. Define a binary operation * on the set A={0,1,2,3,4,5} as a*b=a+b (mod...

    Text Solution

    |

  13. Define a binary operation * on the set A={1,\ 2,\ 3,4} as a*b=a b\ ...

    Text Solution

    |

  14. Construct the composition table for the composition of functions (o...

    Text Solution

    |

  15. Construct the composition table for xx4 on set S={0,\ 1,\ 2,\ 3} .

    Text Solution

    |

  16. Construct the composition table for +5 on set S={0,\ 1,\ 2,\ 3,\ 4}...

    Text Solution

    |

  17. Construct the composition table for xx6 on set S={0,\ 1,\ 2,\ 3,\ 4...

    Text Solution

    |

  18. Construct the composition table for xx5 on Z5={0,\ 1,\ 2,\ 3,\ 4} .

    Text Solution

    |

  19. For the binary operation xx(10) on set S={1,\ 3,\ 7,\ 9} , find the...

    Text Solution

    |

  20. For the binary operation xx7 on the set S={1,\ 2,\ 3,\ 4,\ 5,\ 6} ,...

    Text Solution

    |