Question

The relation $R = \{(4,4), (4,5), (5, 7), (4, 8), (5, 5), (7, 8), (7, 7), (7, 5), (8, 8), (8, 7), (8, 5), (9, 9)\}$ on the set $A=\{4,5,7,8,9\}$ is ________.

• A. transitive
• B. symmetric
• C. reflexive
• D. equivalence relation
• E. a function

Hint:

Reflexive means every element is related to itself.

Answer 1

The Correct Option is C

Step 1: Understanding the Concepts
To solve this problem, we need to check the properties of the given relation R on the set A.
Reflexive: A relation R on a set A is reflexive if for every element \(a \in A\), the pair \((a, a)\) is in R.
Symmetric: A relation R on a set A is symmetric if whenever \((a, b) \in R\), then \((b, a) \in R\).
Transitive: A relation R on a set A is transitive if whenever \((a, b) \in R\) and \((b, c) \in R\), then \((a, c) \in R\).
Equivalence Relation: A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
Function: A relation is a function if for each element in the domain (the first element of the pairs), there is exactly one corresponding element in the codomain (the second element of the pairs).
Step 2: Detailed Explanation
Let's check each property for the given relation R on set A = \{4, 5, 7, 8, 9\}.
1. Checking for Reflexivity:
We need to check if \((a, a) \in R\) for all \(a \in A\).
The elements of A are 4, 5, 7, 8, and 9.
- Is (4, 4) in R? Yes.
- Is (5, 5) in R? Yes.
- Is (7, 7) in R? Yes.
- Is (8, 8) in R? Yes.
- Is (9, 9) in R? Yes.
Since all pairs \((a, a)\) are in R, the relation is reflexive.
2. Checking for Symmetry:
We need to check if for every \((a, b) \in R\), \((b, a)\) is also in R.
Let's find a counterexample. - We see that \((4, 5) \in R\). For R to be symmetric, \((5, 4)\) must also be in R. Looking at the set R, \((5, 4)\) is not present.
Therefore, the relation is not symmetric.
3. Checking for Transitivity:
We need to check if for every \((a, b) \in R\) and \((b, c) \in R\), \((a, c)\) is also in R.
Let's find a counterexample. - We see that \((4, 5) \in R\) and \((5, 7) \in R\). For R to be transitive, \((4, 7)\) must also be in R. Looking at the set R, \((4, 7)\) is not present.
Therefore, the relation is not transitive.
4. Checking for Equivalence Relation:
An equivalence relation must be reflexive, symmetric, and transitive. Since R is not symmetric and not transitive, it is not an equivalence relation.
5. Checking if it is a Function:
For a relation to be a function from A to A, each element of A must map to exactly one element.
- The element 4 maps to 4, 5, and 8 (since (4, 4), (4, 5), and (4, 8) are in R).
Since the element 4 maps to more than one element, the relation is not a function.
Step 3: Final Answer
The only property that the relation R satisfies is reflexivity.