Show that the relation $R$ in the set $\mathbb{R}$ of real numbers defined as $R = \{(a, b) : a \le b\}$ is reflexive and transitive but not symmetric.
Step-by-Step Solution
Reflexive: $a \le a \Rightarrow (a,a) \in R, \forall a \in \mathbb{R}$. [0.5 Mark]
Not Symmetric: $(1,2) \in R$ as $1 \le 2$, but $(2,1)
otin R$ as $2 \le 1$ false. [0.5 Mark]
Transitive: $a \le b, b \le c \Rightarrow a \le c \Rightarrow (a,c) \in R$. Proved! [1.0 Mark]
---
🎯 Official CBSE Marking Scheme:
Proving Reflexivity: 0.5 Mark
Disproving Symmetry with counterexample: 0.5 Mark
Proving Transitivity: 1.0 Mark
Correct Answer: