Question

The relation \( R = \{(1,3), (2,3), (2,4), (3,1), (4,4), (4,1)\} \) on the set \( X = \{1,2,3,4\} \) is:

• A. a 1-1 function
• B. reflexive
• C. transitive
• D. not symmetric
• E. an onto function

Hint:

Check for pairs $(a, b)$ where $(b, a)$ is missing to quickly disprove symmetry.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to analyze the given relation R on the set X and check the properties listed in the options.
The given set is \( X = \{1, 2, 3, 4\} \).
The relation is \( R = \{(1,3), (2,3), (2,4), (3,1), (4,4), (4,1)\} \).
Step 2: Detailed Explanation:
Let's check each option:
(A) a 1-1 function: A relation is a function if each element in the domain maps to exactly one element in the codomain. Here, the element 2 maps to both 3 and 4 (since (2,3) and (2,4) are in R). Also, 4 maps to 4 and 1. Thus, R is not a function, and therefore cannot be a 1-1 function.
(B) reflexive: A relation R on a set X is reflexive if for every element \(x \in X\), the pair \((x, x) \in R\).
For X = \{1, 2, 3, 4\}, we need to have (1,1), (2,2), (3,3), and (4,4) in R.
The relation R contains (4,4), but it is missing (1,1), (2,2), and (3,3). Hence, R is not reflexive.
(C) transitive: A relation R is transitive if for any \((a, b) \in R\) and \((b, c) \in R\), we must also have \((a, c) \in R\).
Let's check for a counterexample. We have \((2, 3) \in R\) and \((3, 1) \in R\). For transitivity, we would need \((2, 1) \in R\). However, (2,1) is not in R. Therefore, the relation is not transitive.
(D) not symmetric: A relation R is symmetric if for every \((a, b) \in R\), the pair \((b, a)\) must also be in R. The relation is not symmetric if we can find at least one pair (a,b) in R such that (b,a) is not in R.
Let's check the pairs:
- \((1,3) \in R\) and \((3,1) \in R\). This pair satisfies symmetry.
- \((2,3) \in R\), but \((3,2)\) is not in R.
Since we found a counterexample, the relation is not symmetric. This statement is true.
(E) an onto function: As established in point (A), R is not a function, so it cannot be an onto function.
Step 3: Final Answer:
Based on the analysis, the only true statement among the options is that the relation is not symmetric.