If R denotes a relation on the set of all ordered pairs of positive integers by (a, b)R(c, d) iff ad show that 'R' is an equivalence relation.

To prove that the relation \( R \) on the set of ordered pairs of positive integers is an equivalence relation, we need to demonstrate that \( R \) satisfies the following three properties of equivalence relations:

1. **Reflexivity**
2. **Symmetry**
3. **Transitivity**

### Given:
Let \( R \) be a relation on the set of all ordered pairs of positive integers, such that:

\[
(a, b) R (c, d) \text{ if and only if } ad = bc.
\]

We will now check each property:

### 1. Reflexivity:
A relation \( R \) is reflexive if for every element \( (a, b) \), it holds that \( (a, b) R (a, b) \).

To check reflexivity, we need to verify that \( (a, b) R (a, b) \), i.e.,

\[
ab = ab.
\]

This is clearly true because any number is always equal to itself. Therefore, \( R \) is **reflexive**.

### 2. Symmetry:
A relation \( R \) is symmetric if whenever \( (a, b) R (c, d) \), then \( (c, d) R (a, b) \).

Suppose \( (a, b) R (c, d) \). By the definition of \( R \), this means that:

\[
ad = bc.
\]

Now, we need to check if \( (c, d) R (a, b) \), i.e., whether \( cb = da \).

Since \( ad = bc \) is equivalent to \( cb = da \) (just rewriting the equality), it follows that \( (c, d) R (a, b) \). Therefore, \( R \) is **symmetric**.

### 3. Transitivity:
A relation \( R \) is transitive if whenever \( (a, b) R (c, d) \) and \( (c, d) R (e, f) \), then \( (a, b) R (e, f) \).

Suppose \( (a, b) R (c, d) \) and \( (c, d) R (e, f) \). This means:

\[
ad = bc \quad \text{and} \quad cf = de.
\]

We need to show that \( (a, b) R (e, f) \), i.e., that \( af = be \).

Multiplying the two equalities \( ad = bc \) and \( cf = de \), we get:

\[
(ad)(cf) = (bc)(de).
\]

Simplifying both sides:

\[
acdf = bcde.
\]

We can cancel \( cd \) from both sides (since \( c \) and \( d \) are positive integers), leaving:

\[
af = be.
\]

Thus, \( (a, b) R (e, f) \), so the relation is **transitive**.

### Conclusion:
Since \( R \) is reflexive, symmetric, and transitive, \( R \) is an **equivalence relation** on the set of ordered pairs of positive integers.

---

Would you like more details, or do you have any questions?

Here are five related questions to deepen your understanding:
1. How would the relation change if we used subtraction instead of multiplication?
2. Can you provide an example of a set and a relation that is not symmetric?
3. How does the concept of equivalence relations relate to partitioning sets?
4. What are some real-life applications of equivalence relations?
5. Can we define equivalence relations on sets that include non-integer elements?

**Tip:** To show transitivity, it's often useful to multiply or combine given equalities to form new ones that satisfy the condition.

If R denotes a relation on the set of all ordered pairs of positive integers by (a, b)R(c, d) iff ad = bc, show that 'R' is an equivalence relation.

Learn how to prove that a relation on ordered pairs of positive integers, defined by (a, b)R(c, d) if ad = bc, is an equivalence relation. This detailed solution explores reflexivity, symmetry, and transitivity, with key algebraic steps. Suitable for high school students in Grades 10-12.

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