The minimum number of elements that must be added to the relation R = {(a, b), (b, c)} on the set {a

Q: Wait, why do we need self-loops like (a, a)? They weren't in the original set!

That is a fantastic observation! Transitivity requires that if (a, b) is in R and (b, a) is in R, then (a, a) must also be in R. Since we added (b, a) to satisfy symmetry, we are forced by the transitive property to include the self-loop (a, a). It is a domino effect where satisfying one property triggers the need for another.

Q: Is there a faster way to find the answer without listing every single pair?

Yes, there is! Think about the final state. To be both symmetric and transitive, the relation must be an equivalence relation. The smallest equivalence relation containing {(a, b), (b, c)} is the universal relation on the set {a, b, c}, which contains all 3^2 = 9 possible pairs. Since we started with 2 pairs, we need to add 9 - 2 = 7 more.

Q: Why did we add (c, a) after adding (a, c)?

Because the relation must be symmetric! Once the transitive property forced us to add the shortcut (a, c), the symmetry property immediately demanded the return path (c, a). In a symmetric relation, you cannot have a one-way street; if you can go from c to a, you must be able to go from a to c.

Question

The minimum number of elements that must be added to the relation $R = \{(a, b), (b, c)\}$ on the set $\{a, b, c\}$ so that it becomes symmetric and transitive is:

Options:

(A) 4

(B) 7

(C) 5

(D) 3

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a map with three islands: $a$, $b$, and $c$. The initial relation $R = \{(a, b), (b, c)\}$ provides two one-way bridges.

Our mission is to transform this network into a perfectly balanced system that is both **symmetric** and **transitive**. This process reveals the structural integrity of mathematical logic.

### Phase 1: The Law of the Return Journey

Symmetry is the law of the return journey. If you can travel from $x$ to $y$, you must be able to return from $y$ to $x$.

Looking at our current bridges, we have a path from $a$ to $b$, so we are obligated to build a return bridge from $b$ to $a$. Similarly, the path from $b$ to $c$ demands a return path from $c$ to $b$.

By adding these two pairs, $\{(b, a), (c, b)\}$, we satisfy the symmetry requirement for our initial set. Our relation now looks like:
$$R_{sym} = \{(a, b), (b, c), (b, a), (c, b)\}$$

### Phase 2: The Transitivity Shortcut

Now, we must address transitivity, which is the law of the express route. It states that if you can travel from $x$ to $y$ and then from $y$ to $z$, there must be a direct shortcut from $x$ to $z$.

Look at our map: we have a path from $a$ to $b$ and a path from $b$ to $c$. This chain forces us to build a direct bridge from $a$ to $c$.

Because we added the pair $(a, c)$, we must ensure our network remains symmetric. Therefore, we are required to add the return bridge $(c, a)$.

### Phase 3: The Chain Reaction of Self-Loops

This is where the logic deepens. Transitivity also applies to round trips.

If we can go from $a$ to $b$ and back from $b$ to $a$, transitivity dictates that we must have a path from $a$ to $a$. This is the birth of the **self-loop**.

The same logic applies to $b$ and $c$. We must add $(a, a)$, $(b, b)$, and $(c, c)$ to complete the system.

### Final Calculation

When we tally everything up, we added:
*   Two pairs for symmetry: $\{(b, a), (c, b)\}$
*   One pair for the transitive shortcut: $(a, c)$
*   One pair for the symmetry of that shortcut: $(c, a)$
*   Three self-loops: $\{(a, a), (b, b), (c, c)\}$

The total number of new elements added is:
$$2 + 1 + 1 + 3 = 7$$

By adding these seven elements to our original two, we have created a universal relation containing all $9$ possible pairs. You have successfully constructed an **equivalence relation** from scratch, resulting in a perfectly symmetric and transitive system.

Animated Solution for Mathematics - Sets and Relations: The minimum number of elements that must be added to the relation $R = {(a, b), (b, c)}$ on the set ${a, b, c}$ so that it becomes symmetric and transitive is:

Select Answer:

Visualized Solution

Visualizing the Initial Relation $R$

Defining Symmetry

Adding Symmetric Pairs

Defining Transitivity

Adding the Transitive Bridge $(a, c)$

Restoring Symmetry for $(a, c)$

Transitivity and Self-Loops

Final Tally of Added Elements

Conclusion

The Sigma Insight: Types of Relations

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Final Calculation

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