Relation \( xv = yu \) on Ordered Pairs

📺 Video Explanation

📝 Question

Let \( A \) be the set of ordered pairs of non-zero integers. Define relation \( R \) as:

\[ (x, y) \in R (u, v) \iff xv = yu \]

Show that \( R \) is an equivalence relation.

---

✅ Solution

🔹 Step 1: Reflexive

For reflexive, we need: \[ (x,y) R (x,y) \]

\[ xy = yx \]

This is always true.

✔ Therefore, the relation is Reflexive. :contentReference[oaicite:0]{index=0}

---

🔹 Step 2: Symmetric

Assume: \[ (x,y) R (u,v) \Rightarrow xv = yu \]

Then: \[ vx = uy \]

So, \[ (u,v) R (x,y) \]

✔ Therefore, the relation is Symmetric. :contentReference[oaicite:1]{index=1}

---

🔹 Step 3: Transitive

Assume: \[ (x,y) R (u,v),\ (u,v) R (p,q) \]

\[ xv = yu,\quad uq = vp \]

Multiply: \[ (xv)(uq) = (yu)(vp) \]

\[ xq = yp \]

So, \[ (x,y) R (p,q) \]

✔ Therefore, the relation is Transitive. :contentReference[oaicite:2]{index=2}

---

🎯 Final Answer

✔ Reflexive: Yes
✔ Symmetric: Yes
✔ Transitive: Yes

\[ \therefore R \text{ is an equivalence relation} \]

---

🚀 Exam Insight

• This represents equality of fractions: \( \frac{x}{y} = \frac{u}{v} \)
• Used in rational numbers concept
• Very important example of equivalence relation