Question

Let A = \{1, 2, 3\}. Then, the number of relations containing (1, 2) and (1, 3), which are reflexive and symmetric but not transitive, is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

When asked to find the number of relations with certain properties, start by building the smallest possible relation that includes all the mandatory elements and satisfies the given properties (like reflexivity and symmetry). Then, check if this minimal relation meets the final condition (e.g., not transitive). Finally, see if adding any other allowed elements still keeps the properties valid.

Answer 1

The Correct Option is A

Step 1: Problem Definition:
The objective is to determine the number of relations R on the set A = \{1, 2, 3\} that satisfy the following four criteria: 1. R must include the ordered pairs (1, 2) and (1, 3). 2. R must be reflexive. 3. R must be symmetric. 4. R must not be transitive. Step 3: Detailed Analysis:
Let R represent the relation under consideration. Conditions 1 & 2 (Reflexivity): For R to be reflexive on set A, it must contain all pairs of the form (a, a) where a is an element of A. Consequently, R must contain \{(1, 1), (2, 2), (3, 3)\}. Conditions 1 & 3 (Symmetry): The relation R must contain (1, 2) and (1, 3). Symmetry dictates that if (a, b) is in R, then (b, a) must also be in R. Given (1, 2) $\in$ R, it follows that (2, 1) $\in$ R. Given (1, 3) $\in$ R, it follows that (3, 1) $\in$ R. The minimal relation R satisfying the first three properties is: \[ R_{min} = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)\} \] Condition 4 (Non-transitivity): We now evaluate if $R_{min}$ is transitive. A relation R is transitive if for any pairs (a, b) $\in$ R and (b, c) $\in$ R, it holds that (a, c) $\in$ R. We search for a counterexample within $R_{min}$. Consider the pairs (2, 1) $\in R_{min}$ and (1, 3) $\in R_{min}$. For transitivity to hold, the pair (2, 3) must be present in $R_{min}$. However, (2, 3) $otin R_{min}$. Similarly, consider (3, 1) $\in R_{min}$ and (1, 2) $\in R_{min}$. Transitivity would require (3, 2) $\in R_{min}$. Yet, (3, 2) $otin R_{min}$. The existence of these counterexamples demonstrates that $R_{min}$ is not transitive. Therefore, $R_{min}$ is one valid relation. Let's investigate if other relations exist. The pairs not present in $R_{min}$ are (2, 3) and (3, 2). To maintain symmetry, if either is added, both must be added. Let's consider $R' = R_{min} \cup \{(2, 3), (3, 2)\}$. $R' = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\}$. This relation is the universal relation $A \times A$. The universal relation on any set is always reflexive, symmetric, and transitive. Thus, R' is transitive. The addition of the pair \{(2, 3), (3, 2)\} results in a transitive relation, which violates condition 4. Step 4: Conclusion:
The relation $R_{min}$ is the sole relation that fulfills all specified conditions. Consequently, there is only 1 such relation.