1. ## equivalence class

Let S be the Cartesian coordinate plane RxR and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation and describe the equivalence class E(7,3).

2. Originally Posted by yenbibi
Let S be the Cartesian coordinate plane RxR and define a relation R on S by (a,b)R(c,d) iff a+d=b+c. Verify that R is an equivalence relation
you must show three things. R is reflexive, symmetric and transitive

(a) Reflexive: you must show that $(a,b)R(a,b)$ for all $(a,b) \in \mathbb{R} \times \mathbb{R}$

(b) Symmetric: you must show that $(a,b)R(c,d) \implies (c,d)R(a,b)$ for $(a,b), (c,d) \in \mathbb{R} \times \mathbb{R}$

(c) Transitive: you must show that $\Bigg[ (a,b)R(c,d) \mbox{ and } (c,d)R(e,f) \Bigg] \implies (a,b)R(e,f)$ for $(a,b), (c,d), (e,f) \in \mathbb{R} \times \mathbb{R}$

can you do that?

and describe the equivalence class E(7,3).
what do you mean E(7,3)? you mean the equivalence class that has the point (7,3) in it?

3. Originally Posted by Jhevon

what do you mean E(7,3)? you mean the equivalence class that has the point (7,3) in it?
Yes, and we can write it as $\overline{(7,3)}=[(7,3)].$

4. Originally Posted by Krizalid
Yes, and we can write it as $\overline{(7,3)}=[(7,3)].$
in that case, it is just $[(7,3)] = \{ (a,b) \mid 7 + b = 3 + a \Longleftrightarrow a - b = 4 \}$