Composition of two equivalence relations

[jasper29]

Homework Statement

The question is let E1 and E2 be equivalence relations on set X. A new relation R is defined as the E1 o E2, the composition of the two relations. We must prove or disprove that R is an equivalence relation.

Homework Equations

The Attempt at a Solution

I know that we must prove
1) reflexive - this is easy just E1 = E1
2) symmetric
3)transitive

but I am unsure of how to prove the last two.
Thanks for any help in advance and if you need more information I will try to provide.

 

[haruspex]

I'm unfamiliar with the concept of composition of equivalence relations. Does it mean that if xE₁y and yE₂z then xE₁oE₂z?

 

[jasper29]

Let E1 and E2 be equivalence relations on a non-empty set X. Define a new relationRonXbyxRyifthereexistsaz∈XsuchthatxE1 zandzE2 y. TherelationR is often denoted as E1 ◦ E2 and is called the composition of the relations E1 and E2. Prove or disprove: R is an equivalence relation on X, which in words is that the composition of equivalence relations is an equivalence relation.

This is the rest of the information

 

[haruspex]

Let E1 and E2 be equivalence relations on a non-empty set X. Define a new relation R on X by xRy if there exists a z∈X such that xE1 z and zE2 y.

OK, that's what I guessed.

1) reflexive - this is easy just E1 = E1

I don't understand your proof there. What do you mean by 'E1=E1'? It's not the equivalence of equivalence relations that's at issue.

2) symmetric
3)transitive

Write those last two out in terms of what you would need to prove re E1oE2.