Question

Let  R = {(1, 2), (2, 3), (3, 3)}} be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is:

• A. \( 10 \)
• B. \( 8 \)
• C. \( 9 \)
• D. \( 7 \)

Hint:

Notice that R links 1 to 2 and 2 to 3 in a chain. Reflexivity, symmetry, and transitivity together force any such chain into one full equivalence class, meaning every possible pair among those elements, including each element with itself, must eventually appear. Work out which elements get forced together, count how many total pairs a complete class of that size needs, then subtract the pairs R already has.

Answer 1

The Correct Option is D

To make a relation \(R\) on a set an equivalence relation, \(R\) must satisfy three properties: reflexivity, symmetry, and transitivity. Let's evaluate the given relation \(R = \{(1, 2), (2, 3), (3, 3)\}\) on the set \(\{1, 2, 3, 4\}\) step-by-step: 

1. Reflexivity: A relation is reflexive if every element is related to itself. Thus, for reflexivity on the set \(\{1, 2, 3, 4\}\), \((1, 1), (2, 2), (3, 3), (4, 4)\) must be in \(R\). Currently, only \((3, 3)\) is in \(R\). Therefore, we need to add:
   • \((1, 1)\)
   • \((2, 2)\)
   • \((4, 4)\)
2. Symmetry: A relation is symmetric if for every \((a, b) \in R\), \((b, a)\) should also be in \(R\). Analyzing the current relation:
   • \((1, 2) \Rightarrow (2, 1)\) should be added.
   • \((2, 3) \Rightarrow (3, 2)\) should be added.
   • \((3, 3)\) is symmetric by itself, so no change is needed.
3. Transitivity: A relation is transitive if for any \((a, b) \in R\) and \((b, c) \in R\), the pair \((a, c)\) should also be in \(R\). Analyze:
   • Given \(, 2)\) and \( (2,\), we need to add \((1, 3)\) for transitivity.

Summarizing the elements needed:

• Reflexivity: \((1, 1), (2, 2), (4, 4)\) → 3 elements
• Symmetry: \((2, 1), (3, 2)\) → 2 elements
• Transitivity: \((1, 3)\) → 1 element

Total elements to be added: \(3 + 2 + 1 = 6\). However, we mistakenly left out the need for reflexive connection for (1, 2) which will further require:

• To ensure hierarchical reflexivity for \((1, 2)\) symmetrically: \((3,1)\), making the total 7 requirements.

Therefore, the minimum number of elements to add so that \(R\) becomes an equivalence relation is 7.

Answer 2

To make the relation \( R = \{(1, 2), (2, 3), (3, 3)\} \) an equivalence relation on the set \( \{1, 2, 3, 4\} \), we need to ensure it satisfies three properties: reflexivity, symmetry, and transitivity.

1. Reflexivity: Each element must be related to itself. Therefore, we must add the pairs: \((1,1)\), \((2,2)\), \((4,4)\).

2. Symmetry: If \((a, b)\) is in the relation, then \((b, a)\) must also be in it. For existing pairs, add: \((2,1)\), \((3,2)\).

3. Transitivity: If \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must also be in it. Evaluate existing pairs:

• \((1,2)\) and \((2,3)\) imply \((1,3)\)
• Using the new pair \((3,2)\) and \((2,1)\), infer \((3,1)\)

Now, enumerating all added pairs, we find: \((1,1)\), \((2,2)\), \((4,4)\), \((2,1)\), \((3,2)\), \((1,3)\), \((3,1)\). Therefore, 7 elements are added in total.

Conclusion: The minimum number of elements to be added is \(7\).

Answer 3

Concept:

• If two elements are linked directly or through a chain of pairs in $R$, then reflexivity, symmetry, and transitivity together force them into one single equivalence class, and every ordered pair within that class, including each element paired with itself, must be present.
• A complete equivalence class of size $n$ needs exactly $n^2$ ordered pairs. The minimum number of pairs to add equals the total pairs required across all forced classes, minus the pairs already present in $R$.

Step 1: Identify how the given pairs link the elements into classes.
$R$ contains $(1,2)$ and $(2,3)$, so $1$, $2$, and $3$ are all linked in a chain and must belong to the same equivalence class.
The element $4$ does not appear in any pair of $R$, so it forms its own separate class $\{4\}$.

Step 2: Count the ordered pairs a complete class of each size requires.
Class $\{1,2,3\}$ has size $3$, so it requires $3^2=9$ ordered pairs: $(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)$.
Class $\{4\}$ has size $1$, so it requires $1^2=1$ ordered pair: $(4,4)$.
Total ordered pairs required $=9+1=10$.

Step 3: Count how many of these required pairs are already in $R$.
$R=\{(1,2),(2,3),(3,3)\}$, and all three of these pairs are among the $9$ pairs required for the class $\{1,2,3\}$.
None of the pairs required for the class $\{4\}$ are present in $R$.
Pairs already present $=3$.

Step 4: Subtract to get the minimum number of pairs to add.
Minimum pairs to add $=10-3=7$.

Final Answer: $7$