Question

In the set $\{1, 2, 3, 4, 5, 6\}$, the relation $R$ defined by $R = \{(a, b) : b = a+1\}$ will be:

• A. not reflexive and symmetric, but transitive
• B. not reflexive and transitive, but symmetric
• C. not reflexive, not symmetric and not transitive
• D. not symmetric and transitive, but reflexive

Hint:

To check properties of relations: - Reflexive: $(a,a)$ for all $a$ - Symmetric: $(a,b) $\Rightarrow$ (b,a)$ - Transitive: $(a,b)$ and $(b,c) $\Rightarrow$ (a,c)$

Answer 1

The Correct Option is C

Step 1: Check reflexivity.
A relation $R$ is reflexive if $(a,a) \in R$ for all $a$. Here, $R = \{(1,2), (2,3), (3,4), (4,5), (5,6)\}$. No pair is of the form $(a,a)$. Hence, $R$ is **not reflexive**.

Step 2: Check symmetry.
If $(a,b) \in R$, then for symmetry $(b,a) \in R$. Example: $(1,2) \in R$, but $(2,1) \notin R$. So, $R$ is **not symmetric**.

Step 3: Check transitivity.
If $(a,b) \in R$ and $(b,c) \in R$, then $(a,c)$ should be in $R$. Example: $(1,2) \in R$ and $(2,3) \in R$, so $(1,3)$ should be in $R$. But $(1,3) \notin R$. Hence, $R$ is **not transitive**.

Step 4: Conclusion.
The relation $R$ is not reflexive, not symmetric, and not transitive. Thus, the correct answer is (C).