If R is the relation on the set of ordered pairs of positive integers such that (a,b),(c,d)∈R whenever ad=bc, Show that R is an equivalence relation

To show that \( R \) is an equivalence relation on the set of ordered pairs of positive integers, we need to verify that \( R \) satisfies the three properties of equivalence relations: reflexivity, symmetry, and transitivity.

**1. Reflexivity:** 
We need to show that for any \((a, b)\), \((a, b) R (a, b)\).

Consider \((a, b)\) where \(a\) and \(b\) are positive integers. We need to check if \((a, b) R (a, b)\). According to the definition of \(R\), \((a, b) R (c, d)\) if and only if \(ad = bc\). Here, \(a = a\) and \(b = b\), so we need to check if \(ab = ba\).

Clearly, \(ab = ba\) (since multiplication is commutative).

Thus, \((a, b) R (a, b)\) and \(R\) is reflexive.

**2. Symmetry:**
We need to show that if \((a, b) R (c, d)\), then \((c, d) R (a, b)\).

Assume \((a, b) R (c, d)\). By the definition of \(R\), this means that \(ad = bc\). 

To show symmetry, we need to check if \(cd = da\). Given that \(ad = bc\), we can rearrange this equation to see that \(cd = da\).

Thus, \((c, d) R (a, b)\) and \(R\) is symmetric.

**3. Transitivity:**
We need to show that if \((a, b) R (c, d)\) and \((c, d) R (e, f)\), then \((a, b) R (e, f)\).

Assume \((a, b) R (c, d)\) and \((c, d) R (e, f)\). By the definition of \(R\), we have:
\[ ad = bc \quad \text{and} \quad cf = de \]

We need to show that \(af = be\).

Starting from the given:
\[ ad = bc \implies a = \frac{bc}{d} \]
\[ cf = de \implies f = \frac{de}{c} \]

Now, multiply \(a\) and \(f\):
\[ af = \left(\frac{bc}{d}\right) \left(\frac{de}{c}\right) = \frac{bc \cdot de}{dc} = \frac{b \cdot de}{d} = be \]

Thus, \(af = be\) which implies \((a, b) R (e, f)\).

Therefore, \(R\) is transitive.

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 on any of these steps or have any questions?

Here are 8 related questions to expand on this topic:

1. Can you give an example of a set of ordered pairs that form an equivalence class under this relation?
2. How can we use the concept of equivalence classes to partition a set?
3. What are the general properties of equivalence relations?
4. How would you prove that a given relation is not an equivalence relation?
5. What are some real-world examples of equivalence relations?
6. How does the concept of an equivalence relation relate to modular arithmetic?
7. Can you describe the difference between a partial order and an equivalence relation?
8. How can equivalence relations be used in computer science, for example in data classification?

**Tip:** When working with equivalence relations, always verify reflexivity, symmetry, and transitivity to confirm that the relation holds across the entire set.

Explore how the relation R on the set of ordered pairs of positive integers forms an equivalence relation. Learn about reflexivity, symmetry, and transitivity with detailed examples and proofs.

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