# Confusion regarding the Quotient Set

Supposing I have a set A and binary relation R such that

$A = \{1,2,3,4,5\}$ $R = \{(1,1), (1,3), (1,4), (2,2), (2,5), (3,1), (3,3), (3,4), (4,1), (4,3), (4,4), (5,2), (5,5)\}$

I want to find the Quotient Set $A/R$, but I'm not totally sure I understood my professor's explanation of how to do so.

From what I gathered, the Quotient Set is the set of all equivalence classes of A under the relation R. Then if R is an equivalence relation, the equivalence class of some element $a$ belonging to $A$ is the set of all elements related to $a$. In that case, would the quotient set of A and R be

$$A/R = \{\{1,3,4\}, \{2,5\}\}$$

Or would I have to include binary pairs in the quotient set, i.e. $A/R = \{\{(1,1), (1,3) ... \}\}$ since the elements of R are binary pairs?

• Your first version is correct: $A/R=\big\{\{1,3,4\},\{2,5\}\big\}$. – Brian M. Scott Oct 22 '16 at 19:37