Question

Let \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 2), (2, 3), (1, 4)\} \) be a relation on \( A \).Let \( S \) be the equivalence relation on \( A \) such that \( R \subseteq S \) and the number of elements in \( S \) is \( n \). Then, the minimum value of \( n \) is _____

Answer 1

Correct Answer: 16

Given \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 2), (2, 3), (1, 4)\} \), for \( R \) to be an equivalence relation, it must satisfy the following properties: reflexive, symmetric, and transitive.

Reflexivity: Add all pairs of the form \((a, a)\), where \( a \in A \):
\(\{(1, 1), (2, 2), (3, 3), (4, 4)\}\)

Symmetry: Add pairs such that if \((a, b) \in R\), then \((b, a)\) must also belong to \( R \):
\(\{(2, 1), (3, 2), (4, 1)\}\)

Transitivity: Ensure that if \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\). For example:
  \((1, 2), (2, 3) \implies (1, 3)\)
  Applying this to all pairs results in:
 \(\{(1, 3), (3, 1), (2, 4), (4, 2), (4, 3), (3, 4)\}\)

Combining all the above, the final relation \( R \) becomes:
\(R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (1, 4), (4, 1), (1, 3), (3, 1), (2, 4), (4, 2), (4, 3), (3, 4)\}\)

Thus, the total number of elements in \( R \) is 16.

\(\boxed{\text{Answer: } 16.}\)