Transitive relation

• Feb 18th 2009, 09:54 PM
namelessguy
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.
• Feb 19th 2009, 04:02 AM
clic-clac
Hi
1) Check $A=\{\{(a,b),(b,c)\},\{(b,d)\}\}$
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\},\$ $x=\{(a,a),(b,b),(c,c),(a,b),(b,a)\},\ y=\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$