Question

A relation \( R = \{(x, y) : \text{Number of pages in} \, x \text{ and } y \text{ are equal} \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.

Hint:

For a relation to be an equivalence relation, it must satisfy the properties of reflexivity, symmetry, and transitivity.

Answer 1

To prove that \( R \) is an equivalence relation, we need to show that it satisfies three properties: 

1. Reflexivity
2. Symmetry
3. Transitivity

1. Reflexivity:
A relation \( R \) is reflexive if \( (x, x) \in R \) for every \( x \in A \).
In this case, the number of pages in a book is always equal to itself. Thus, for every book \( x \), the relation holds: \[ (x, x) \in R \text{because the number of pages in } x \text{ is equal to itself}. \] Hence, \( R \) is reflexive. 

2. Symmetry:
A relation \( R \) is symmetric if whenever \( (x, y) \in R \), we also have \( (y, x) \in R \).
If the number of pages in book \( x \) is equal to the number of pages in book \( y \), then the number of pages in book \( y \) is also equal to the number of pages in book \( x \). Thus, if \( (x, y) \in R \), we also have \( (y, x) \in R \). Therefore, \( R \) is symmetric. 

3. Transitivity:
A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), we also have \( (x, z) \in R \).
If the number of pages in book \( x \) is equal to the number of pages in book \( y \), and the number of pages in book \( y \) is equal to the number of pages in book \( z \), then the number of pages in book \( x \) is equal to the number of pages in book \( z \). Hence, if \( (x, y) \in R \) and \( (y, z) \in R \), we also have \( (x, z) \in R \). Therefore, \( R \) is transitive. 

Conclusion:
Since the relation \( R \) is reflexive, symmetric, and transitive, we conclude that \( R \) is an equivalence relation.