Home
Class 12
MATHS
Let "T" be the set of all triangles in...

Let `"T"` be the set of all triangles in a plane with `"R"` as relation in `"T"` given by
`"R"={("T"_1,"T"_2):("T")_1 "congruent to" "T"_2}`
. Show that `"R"` is an equivalence relation.

Text Solution

Verified by Experts

`R=[(T_1,T_2):T_1` IS Congruent to` T_2`]
Check reflexive
since every triangle is congruent to itself
Triangle T is congruent to Triangle T
`(T,T)inR`
So R is reflexive
check symmetric
If `T_1`is congruent to `T_2`
then `T_2`is congruent to `T_1`
So, if `(T_1,T_2)inR`,Then `(T_2,T_1)inR`
Hence ,R is symmetric
Check transivity
If `T_1` is congruent to `T_2`,&`T_2` is congruent to `T_3`
then `T_1`is congruent to `T_3`
So ,If `(T_1,T_2),(T_2,T_3)inR`
then `(T_1,T_3)inR`
SO,R is Transitive
Since `R` is reflexive,symmetric & transitive.
Therefore ,R is an equivalence relation
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise EXERCISE 1.4|13 Videos

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise EXERCISE 1.2|12 Videos

  • RELATIONS AND FUNCTIONS

    NCERT|Exercise EXERCISE 1.3|14 Videos

  • PROBABILITY

    NCERT|Exercise EXERCISE 13.2|18 Videos

  • SETS

    NCERT|Exercise EXERCISE 1.3|1 Videos

Similar Questions

Explore conceptually related problems

Let T be the set of all triangles in a plane with R as relation in T given by R={(T_(1),T_(2)):T_(1)~=T_(2)}. Show that R is an equivalence relation.

Let A be the set of all triangles in a plane show that the relation R={(Delta_(1),Delta_(2)):Delta_(1)~Delta_(2)} is an equivalence relation on A .

Show that the relation R defined in the set A of all triangles as R={(T_(1),T_(2)):T_(1) is similar to T_(2) }, is equivalence relation.

Show that the relation R defined on the set A of all triangles in a plane as R={(T_(1),T_(2)):T_(1) is similar to T_(2) ) is an equivalence relation.Consider three right angle triangle T_(1) with sides 3,4,5;T_(2),10. sides 5,12,13 and T_(3) with sides 6,8,10. Which triangles among T_(1),T_(2) and T_(3) are related?

Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as a\ R\ b if a is congruent to b for all a ,\ b in T . Then, R is (a) reflexive but not symmetric (b) transitive but not symmetric (c) equivalence (d) none of these

Show that the relation is congruent to on the set of all triangles in a plane is an equivalence relation

NCERT-RELATIONS AND FUNCTIONS-SOLVED EXAMPLES
  1. Show that the relation R in the set {1, 2, 3}given by R = {(1, 1), (2...

    Text Solution

    |

  2. Show that the relation R on the set Z of integers, given by R={(a ,\ b...

    Text Solution

    |

  3. Let "T" be the set of all triangles in a plane with "R" as relation ...

    Text Solution

    |

  4. Let L be the set of all lines in a plane and R be the relation in L de...

    Text Solution

    |

  5. Let A be the set of all students of a boys school. Show that the rela...

    Text Solution

    |

  6. Show that – a is the inverse of a for the addition operation '+' on R ...

    Text Solution

    |

  7. Show that zero is the identity for addition on R and 1 is the identit...

    Text Solution

    |

  8. Show that the vv: R ->R given by (a , b)->m a x {a , b}and the ^^: R -...

    Text Solution

    |

  9. Let P be the set of all subsets of a given set X. Show that uu: P xx ...

    Text Solution

    |

  10. Show that ∗ : RxxR->R given by (a ,b)->a+4b^2is a binary operation.

    Text Solution

    |

  11. Show that subtraction and division are not binary operations on N.

    Text Solution

    |

  12. Show that *: Rxx R ->Rgiven by a*b = a +2bis not associative.

    Text Solution

    |

  13. Show that addition and multiplication are associative binary operatio...

    Text Solution

    |

  14. Show that *: R xxR ->Rdefined by a*b = a +2bis not commutative.

    Text Solution

    |

  15. Show that + : R xx R ->R and xx : R xx R ->R are commutative binary ...

    Text Solution

    |

  16. Let Y = {n^2: n in N} in N. Consider f : N ->Yas f(n)=n^2. Show tha...

    Text Solution

    |

  17. Let f : N ->R be a function defined as f(x)=4x^2+12 x+15. Show that f...

    Text Solution

    |

  18. Consider f : N ->N, g : N ->Nand h : N ->Rdefined asf (x) = 2x, g (y) ...

    Text Solution

    |

  19. Consider f : {1, 2, 3} ->{a , b , c}and g : {a , b , c} ->{a p p l e ,...

    Text Solution

    |

  20. Consider functions f and g such that composite gof is defined and is ...

    Text Solution

    |