Check whether the relation $R$ in $\mathbb{R}$ defined by $R = \{(a, b) : a \le b^3\}$ is reflexive, symmetric or transitive.
Step-by-Step Solution
Not reflexive: $1/2 \le (1/2)^3 = 1/8$ is false. [0.5 Mark]
Not symmetric: $(1,2) \in R$ as $1 \le 8$, but $(2,1)
otin R$ as $2 \le 1$ false. [0.5 Mark]
Not transitive: Counterexample $(3, 1.5) \in R, (1.5, 1.2) \in R$ but $(3, 1.2)
otin R$. [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Disproving Reflexivity: 0.5 Mark
Disproving Symmetry: 0.5 Mark
Disproving Transitivity: 1.0 Mark
Correct Answer: