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A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilisers 
(
F
)
(F), 245 ponds are polluted by pesticides 
(
P
)
(P) and 257 ponds are polluted by pharmaceutical products 
(
P
h
)
(Ph). 100 ponds are polluted by fertilisers and pesticides, 82 ponds are polluted by fertilisers and pharmaceutical products, 77 ponds are polluted by pesticides and pharmaceutical products.Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilisers and pharmaceutical products. Which of the following is/are true?

Relation is reflexive.

Relation is transitive.

Relation is symmetric.

This is an equivalence relation.

Let's break down the problem by analyzing the provided data and determine whether the defined relation satisfies the conditions for being an equivalence relation.

### Given Data:
- Total ponds = 525
- Ponds polluted by fertilizers (\( F \)) = 230
- Ponds polluted by pesticides (\( P \)) = 245
- Ponds polluted by pharmaceuticals (\( Ph \)) = 257
- Ponds polluted by fertilizers and pesticides = 100
- Ponds polluted by fertilizers and pharmaceuticals = 82
- Ponds polluted by pesticides and pharmaceuticals = 77

### Defining the relation:
The relation \( R \) is defined as follows: two ponds \( A \) and \( B \) are related if both are polluted by fertilizers (\( F \)) and pharmaceuticals (\( Ph \)).

We now need to check whether the relation \( R \) satisfies the following properties to determine if it is an equivalence relation.

1. **Reflexive**: A relation is reflexive if every element is related to itself. In this case, for every pond polluted by both fertilizers and pharmaceuticals, it should be related to itself. Since a pond that is polluted by both fertilizers and pharmaceuticals would naturally be related to itself, the relation is reflexive.

2. **Symmetric**: A relation is symmetric if, for any two ponds \( A \) and \( B \), if \( A \) is related to \( B \), then \( B \) is also related to \( A \). In this case, if pond \( A \) is polluted by both fertilizers and pharmaceuticals, and pond \( B \) is also polluted by both fertilizers and pharmaceuticals, then they are related to each other. This condition is symmetric by definition, as the relation is mutual.

3. **Transitive**: A relation is transitive if, for any ponds \( A \), \( B \), and \( C \), if \( A \) is related to \( B \), and \( B \) is related to \( C \), then \( A \) must be related to \( C \). In this case, if pond \( A \) and pond \( B \) are both polluted by fertilizers and pharmaceuticals, and pond \( B \) and pond \( C \) are also polluted by fertilizers and pharmaceuticals, then all three ponds are polluted by fertilizers and pharmaceuticals. Thus, the relation is transitive.

4. **Equivalence relation**: A relation is an equivalence relation if it is reflexive, symmetric, and transitive. Since this relation satisfies all three properties (reflexivity, symmetry, and transitivity), the relation \( R \) is indeed an equivalence relation.

### Conclusion:
The relation defined on the set of ponds that are polluted by both fertilizers and pharmaceuticals is **reflexive, symmetric, and transitive**, making it an equivalence relation.

Would you like to dive deeper into any part of the explanation or have any questions?

### 5 Related Questions:
1. How can we represent the given relations between the polluted ponds using set notation?
2. What is the intersection of sets \( F \) and \( Ph \) in this problem, and how many ponds does it represent?
3. How would the Venn diagram for this situation look if all polluted categories are considered?
4. Can a relation be reflexive but not symmetric or transitive? Provide examples.
5. How do you verify whether a relation is an equivalence relation in other contexts, such as geometry or algebra?

### Tip:
When verifying equivalence relations, always check reflexivity, symmetry, and transitivity individually. If all three are satisfied, the relation is an equivalence relation.

A survey was conducted on pollution of 525 ponds across some cities. It was found that 230 ponds are polluted by fertilizers, 245 ponds are polluted by pesticides, and 257 ponds are polluted by pharmaceutical products. 100 ponds are polluted by fertilizers and pesticides, 82 ponds are polluted by fertilizers and pharmaceutical products, and 77 ponds are polluted by pesticides and pharmaceutical products. Define a relation on the set of 525 ponds such that two ponds are related if both are polluted by fertilizers and pharmaceutical products. Which of the following is/are true? Relation is reflexive. Relation is transitive. Relation is symmetric. This is an equivalence relation.

Explore the concept of equivalence relations through a detailed analysis of a survey on 525 ponds polluted by fertilizers, pesticides, and pharmaceutical products. Learn how set theory applies to real-world data and understand the properties of reflexivity, symmetry, and transitivity in defining relations.

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