Home
Class 12
MATHS
Let Z be the set of integers. Show th...

Let `Z` be the set of integers. Show that the relation `R={(a , b): a , bZ` and `a+b` is even} is an equivalence relation on Z.

Text Solution

Verified by Experts

`R = {(a,b): a,b in Z and a+b` is even`}`
`a+a = 2a`, which is even. `:. (a,a) in R`
`:. R` is reflexive.

`(a,b) in R`.
It means `a+b` is even.
...
Promotional Banner

Topper's Solved these Questions

  • PROBABILITY

    RD SHARMA|Exercise Solved Examples And Exercises|422 Videos

  • SCALAR OR DOT PRODUCT

    RD SHARMA|Exercise Solved Examples And Exercises|232 Videos

Similar Questions

Explore conceptually related problems

Let Z be the set of integers.Show that the relation R={(a,b):a,b in Z and a+b is even } is an equivalence relation on Z .

If Z is the set of integers. Then, the relation R={(a,b):1+ab gt 0} on Z is

Show that the relation R defined by R={(a,b):a-b is divisible by 3;a,b in Z} is an equivalence relation.

Let R={(a,b):a,b inZ" and "(a-b)" is even"}. Then, show that R is an equivalence relation on Z.

Let R={(a,b):a,b in Z and (a-b) is divisible by 5}. Show that R is an equivalence relation on Z.

Prove that the relation R on Z defined by (a,b)in R hArr a-b is divisible by 5 is an equivalence relation on Z .

Show that the relation R in the set A={1,2,3,4,5} given by R={(a,b):|a-b| is even }, is an equivalence relation.

Statement-1: On the set Z of all odd integers relation R defined by (a, b) in R iff a-b is even for all a, b in Z is an equivalence relation. Statement-2: If a relation R on a set A is symmetric and transitive, then it is reflexive and hence an equivalence relation, because (a, b) in Rimplies(b, a)in R" [By symmetry]" (a, b)in R and (b, a) in Rimplies (a,a)in R " [By transitivity]"

Let S be the set of all rest numbers and lets R={(a,b):a,bin S and a=+-b}. Show that R is an equivalence relation on S.

RD SHARMA-RELATIONS-Solved Examples And Exercises
  1. Let Z be the set of all integers and Z0 be the set of all non=ze...

    Text Solution

    |

  2. Prove that every identity relation on a set is reflexive, but the c...

    Text Solution

    |

  3. Let Z be the set of integers. Show that the relation R={(a , b): a ...

    Text Solution

    |

  4. If R is an equivalence relation on a set A, then R^-1 is A. reflexiv...

    Text Solution

    |

  5. On the set N of all natural numbers, a relation R is defined as follow...

    Text Solution

    |

  6. If R1 and R2 are equivalence relations in a set A, show that R1nnR2...

    Text Solution

    |

  7. Let Z be the set of all integers and Z0 be the set of all non=zero...

    Text Solution

    |

  8. Let R be the equivalence relation in the set A={0,1,2,3,4,5} given ...

    Text Solution

    |

  9. An integer m is said to be related to another integer n if m is a mult...

    Text Solution

    |

  10. Show that the relation geq on the set R of all real numbers is r...

    Text Solution

    |

  11. m is said to be related to n if m and n are integers and m-n is divisi...

    Text Solution

    |

  12. Let O be the origin. We define a relation between two points P and ...

    Text Solution

    |

  13. Show that the relation R defined by R={(a , b):a-b is divisible ...

    Text Solution

    |

  14. Prove that a relation R on a set A is symmetric iff R=R^-1

    Text Solution

    |

  15. Three relations R1, R2 and R3 are defined on set A={a , b , c} as foll...

    Text Solution

    |

  16. Let a relation R1 on the set R of real numbers be defined as (a , b) ...

    Text Solution

    |

  17. Let S be the set of all points in a plane and R be a relation on S def...

    Text Solution

    |

  18. The following relations are defined on the set of real number: a ...

    Text Solution

    |

  19. Prove that every identity relation on a set is reflexive, but the c...

    Text Solution

    |

  20. Let R be a relation defined on the set of natural numbers N as R={(...

    Text Solution

    |