Results 1 to 2 of 2

Math Help - Transitive relation

  1. #1
    Member
    Joined
    Nov 2007
    Posts
    108

    Transitive relation

    1)Prove or disprove: If A is a set for which each R \in A is a transitive relation, then \bigcup A is a transitive relation.

    I think this is true. So I tried to prove it. Let (x,y) \in \bigcup A and (y,z) \in \bigcup A. Then by definition of union, there exists R such that (x,y) \in R \in A and (y,z) \in R \in  A. Since R is transitive by definition of set A, we have (x,z) \in R \in A. Hence, (x,z) \in \bigcup A
    I am not sure about my argument here. Is it possible that (x,y) \in R \in A while (y,z) \in R' \in A?Any help is appreciated.

    2) Prove or disprove: If A is a nonempty set each of whose elements is an equivalence relation on the set B, then \bigcup A is an equivalence relation on B.
    I don't think this is correct, but I can't come up with a counterexample. I tried the problem but with \bigcap A, and it seems that three properties of equivalence relation do hold for this case, but for union, somehow I don't see it follows.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi
    1) Check A=\{\{(a,b),(b,c)\},\{(b,d)\}\}
    Then by definition of union, there exists such that and
    Take care, as you said after, the "first R" has no to be equal to the "second one"!

    2) Here again, think of a particular but simple case where A has only two elements. What if there is one class in the first relation of equivalence intersects partially a class of the second relation, for instance: (the idea is to lose the transitivity)

    B=\{a,b,c\},\ A=\{x,y\},\ x=\{(a,a),(b,b),(c,c),(a,b),(b,a)\},\ y=\{(a,a),(b,b),(c,c),(b,c),(c,b)\}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Transitive Relation
    Posted in the Discrete Math Forum
    Replies: 6
    Last Post: November 13th 2011, 02:40 PM
  2. Determine whether the following relation is transitive
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: May 4th 2010, 11:25 AM
  3. [SOLVED] Transitive relation help please?
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: February 23rd 2010, 01:39 AM
  4. Transitive relation help
    Posted in the Discrete Math Forum
    Replies: 8
    Last Post: December 3rd 2009, 04:40 PM
  5. Transitive closure relation.
    Posted in the Discrete Math Forum
    Replies: 9
    Last Post: March 8th 2009, 04:38 AM

Search Tags


/mathhelpforum @mathhelpforum