Question

A survey has been done to know the population density of a city. To make the survey efficient, the required data has been collected with the help of set S. Any element (a, b) belongs to set S if and only if a and b reside in the same colony. Examine whether S is an equivalence relation or not.

Options:
a. Set S is an equivalence relation.
b. Set S is not an equivalence relation.
c. The information is insufficient to determine.
d. The concept of equivalence relations is not applicable here.

Answer 1

Set S is an equivalence relation because it satisfies all three necessary conditions: reflexivity, symmetry, and transitivity, as demonstrated by examining relationships within the same colony.

Step-by-step explanation:

To determine whether the set S forms an equivalence relation, we must check if it satisfies three properties: reflexivity, symmetry, and transitivity.

• Reflexivity: An element (a,b) belongs to set S if a and b reside in the same colony, so every element (a,a) will be part of the set because any individual resides in the same colony as themselves. This satisfies the reflexivity condition.
• Symmetry: If (a,b) is in S, then a and b are from the same colony; thus, (b,a) should also be in S since b and a also reside in the same colony. This satisfies the symmetry condition.
• Transitivity: If (a,b) and (b,c) are in S, then a, b, and c all reside in the same colony, and so (a,c) should also be in S. This satisfies the transitivity condition.

Since set S meets all three conditions (reflexivity, symmetry, and transitivity), it is an equivalence relation. Therefore, the correct answer is: Set S is an equivalence relation.