Question

Let $X=\{1,2,3,4,5,6,7}$. Let $R=\{(1,1), (1, 2), (1, 3), (5, 6), (6, 7), (7,5)\}$ be a relation on $X$. Then the relation $R$ will become reflexive if we include the pairs

• A. (2,7), (3,3), (5,5), (6,6) and (7,7)
• B. (2,2), (4,3), (5,5), (6,6) and (7,7)
• C. (2,2), (3,3), (5,5), (6,6) and (7,6)
• D. (2,2), (3,3), (5,5) and (7,7)
• E. (2,2), (3,3), (4,4), (5,5), (6,6) and (7,7)

Hint:

Logic Tip: Always remember that for reflexivity, the condition $(a,a) \in R$ must hold true for \textbf{every single element} present in the parent set. Missing even one identity pair makes it non-reflexive.

Answer 1

Answer: (E)

Concept:
Relation and Functions - Reflexive Relation.
Step 1: Recall the definition of a reflexive relation.
A relation $R$ on a set $X$ is said to be reflexive if every element in $X$ is related to itself.
Mathematically, $(a, a) \in R$ for all $a \in X$.
Step 2: Identify the required pairs for the given set X.
Given the set $X = \{1, 2, 3, 4, 5, 6, 7\}$.
For $R$ to be reflexive, it MUST contain the following pairs: $$ \{(1,1), (2,2), (3,3), (4,4), (5,5), (6,6), (7,7)\} $$
Step 3: Compare required pairs with the given relation R.
The given relation is $R = \{(1,1), (1, 2), (1, 3), (5, 6), (6, 7), (7,5)\}$.
We can see that the pair $(1,1)$ is already present in $R$.
Step 4: Determine the missing pairs.
The pairs needed to satisfy the reflexive condition that are currently missing are: $$ (2,2), (3,3), (4,4), (5,5), (6,6), \text{ and } (7,7) $$ Including these specific pairs will make the relation $R$ reflexive.