# Thread: equivalence relation

1. ## equivalence relation

question

An equivalence relation R defined on a set contains
the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R A x A.

have to do this for a assignment but have no idea where to start I missed the class covering this. could anyone tell me the basic technique how to solve this so i know where to start at least!

2. ## Re: equivalence relation

An equivalence relation on A is a subset of A x A with certain properties. The three pairs you show do not currently satisfy those properties.

1. You need reflexivity: For all $a \in A,(a,a) \in R$.
2. You need symmetry: If $(a,b) \in R$, then $(b,a) \in R$.
3. You need transitivity: If $(a,b)\in R$ and $(b,c) \in R$ then $(a,c) \in R$.

3. ## Re: equivalence relation

sorry wrote question wrong

An equivalence relation R defined on a set A ={1,2,3,4}contains
the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R ≠ A x A.

4. ## Re: equivalence relation

What I am saying is, add pairs until you satisfy those three properties.

5. ## Re: equivalence relation

im sorry i dont understand

6. ## Re: equivalence relation

Check: for any $a \in A$, do you have $(a,a) \in R$? No. So, add those pairs that you are missing. $(1,2) \in R$. Do you have $(2,1) \in R$? No. So add that pair. Etc.

Edit: By add those pairs, I mean add the pairs to the list of pairs you know must be in $R$. You started with the three given pairs. Then, from those three, you use the properties to figure out which other pairs must also be in $R$. In the end, you should find 10 pairs in $R$. If another equivalence relation $R'$ has $R \subseteq R'$ and you know $R' \setminus R \neq \emptyset$, then it is easy to show that $R' = A \times A$. In other words, $R$ is the unique equivalence relation on $A$ with those three pairs.

7. ## Re: equivalence relation

Originally Posted by ronanbrowne88
question
An equivalence relation R defined on a set contains
the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R A x A.
have to do this for a assignment but have no idea where to start I missed the class covering this. could anyone tell me the basic technique how to solve this so i know where to start at least!
Define $F=\{(1,1),(1,2),(2,3)\}$. The diagonal is $\Delta_A=\{(1,1),(2,2),(3,3),(4,4)\}$

Define $G=F\cup\Delta_A$ and $H=(G\circ G)\cup G$

Define $R=H\cup H^{-1}$.

Show that $R$ is an equivalence relation.