Let's break this down step by step, starting with the definition of the relation.

Problem Restatement:

We have a set of 525 ponds, and a relation is defined on this set where two ponds are related if both are polluted by fertilisers and pharmaceutical products.

Step 1: Check if the Relation is Reflexive

A relation $R$ is reflexive if, for every element $x$ in the set, $xRx$ holds (i.e., every pond is related to itself).

• If a pond is polluted by both fertilisers and pharmaceutical products, it is related to itself because it meets the condition of being polluted by both substances.

• Thus, if a pond is polluted by both fertilisers and pharmaceutical products, the relation is reflexive for those ponds.

  Therefore, the relation is reflexive for any pond that meets the pollution condition.

Step 2: Check if the Relation is Symmetric

A relation $R$ is symmetric if, whenever $xRy$ holds, then $yRx$ must also hold.

• In this case, if two ponds (say $P_1$ and $P_2$) are polluted by both fertilisers and pharmaceutical products, then:

  • $P_1$ is related to $P_2$ because they both are polluted by these products.
  • Symmetry holds because $P_2$ is also related to $P_1$ for the same reason.

  Therefore, the relation is symmetric.

Step 3: Check if the Relation is Transitive

A relation $R$ is transitive if, whenever $xRy$ and $yRz$, then $xRz$ must also hold.

• Suppose $P_1$ is related to $P_2$ (both polluted by fertilisers and pharmaceutical products), and $P_2$ is related to $P_3$ (both also polluted by these products).
• If $P_1$ and $P_3$ are both polluted by fertilisers and pharmaceutical products, then $P_1$ is related to $P_3$, satisfying transitivity.

Therefore, the relation is transitive.

Step 4: Is this an Equivalence Relation?

An equivalence relation must satisfy three properties:

• Reflexivity
• Symmetry
• Transitivity

Since the relation is reflexive, symmetric, and transitive, this is an equivalence relation for ponds that are polluted by both fertilisers and pharmaceutical products.

Conclusion:

• The relation is reflexive.
• The relation is symmetric.
• The relation is transitive.
• This is an equivalence relation.

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Do you want more details or have any further questions?

Here are some related questions:

1. What is an example of a non-reflexive relation?
2. Can a relation be symmetric but not transitive? Give an example.
3. What are the implications of a relation being an equivalence relation in practical applications?
4. How would the relation change if we only consider ponds polluted by fertilisers but not pharmaceuticals?
5. How can we visualize equivalence relations in terms of partitioning sets?

Tip: An equivalence relation on a set partitions the set into disjoint subsets, where all elements in a subset are related to each other.