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

To solve the problem, we need to construct an equivalence relation \( S \) on set \( A = \{1, 2, 3, 4\} \) where \( R = \{(1, 2), (2, 3), (1, 4)\} \) is a subset of \( S \). An equivalence relation must be reflexive, symmetric, and transitive.

Step 1: Reflexivity
For reflexivity, every element in \( A \) must relate to itself. Therefore, we add \((1,1)\), \((2,2)\), \((3,3)\), and \((4,4)\) to \( S \).

Step 2: Symmetry 
For symmetry, if \((a,b)\in S\), then \((b,a)\) must also be in \( S \). Using \( R \), we add the pairs \((2,1)\), \((3,2)\), and \((4,1)\).

Step 3: Transitivity
For transitivity, if \((a,b)\in S\) and \((b,c)\in S\), then \((a,c)\) must be in \( S \). We now apply transitivity for the relations:

• \((1,2)\) and \((2,3)\) imply \((1,3)\) and \((3,1)\).
• \((1,2)\) and \((2,1)\) imply \((1,1)\), already included as part of reflexivity.
• \((2,3)\) and \((3,2)\) imply \((2,2)\), already included.
• \((1,4)\) and \((4,1)\) imply \((1,1)\), already included.
• \((3,1)\) and \((1,4)\) imply \((3,4)\) and \((4,3)\).

By following reflexivity, symmetry, and transitivity, we find that the smallest equivalence relation \( S \) is \( \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2), (1,4), (4,1), (1,3), (3,1), (3,4), (4,3), (4,2), (2,4)\} \).

Thus, the minimal \( S \) contains \( 16 \) elements. Confirming this value fits within the given range: (16, 16).

The minimum value of \( n \) is \( 16 \).

Answer 2

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.

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