Q.Check whether the relation R in $\mathbf{R}$ defined by $R = \{(a, b) : a \le b^3\}$ is reflexive, symmetric or transitive.

The key idea is to test each property individually — reflexivity, symmetry, transitivity — using the definition $a \le b^3$.

1. Reflexive: For any $a \in \mathbf{R}$, we need $a \le a^3$. This fails for $a = \frac12$, since $\frac12 \le \frac18$ is false. So $R$ is not reflexive.

2. Symmetric: If $a \le b^3$, does $b \le a^3$ follow? Take $a=1$, $b=2$: $1 \le 8$ holds, but $2 \le 1$ is false. So $R$ is not symmetric. …

We test each property directly against the defining rule $a \le b^3$. A single counterexample is enough to disprove a property.

1. Reflexivity

Reflexivity would require $a \le a^3$ for every real number $a$.

Take $a = \tfrac{1}{2}$:

This is false, since $\tfrac{1}{2} > \tfrac{1}{8}$. So $\left(\tfrac{1}{2}, \tfrac{1}{2}\right) \notin R$.

$R$ is not reflexive.

2. Symmetry

Symmetry would require: whenever $a \le b^3$, also $b \le a^3$.

Take $a = 1,\ b = 2$:

But

$R$ is not symmetric.

3. Transitivity …

Method: Testing an Inequality Relation with a Cube

Use this for relations like $a\,R\,b \iff a\le b^3$. The cube on one side is exactly what makes each property fail, so aim your test cases at that asymmetry.

Steps

Step 1: Reflexive — check $a\le a^3$ across the domain

Try values in $(0,1)$ where a cube shrinks the number: $a=\tfrac12$ gives $\tfrac12\le\tfrac18$, false, so it is not reflexive.

Step 2: Symmetric — swap a mismatched pair

Pick $a$ small and $b$ larger: $(1,2)$ has $1\le 8$ true, but $(2,1)$ needs $2\le 1$, false.

Step 3: Transitive — chain toward a broken bound …

Common Mistakes

Mistake 1: Testing reflexivity only with numbers $>1$

Why it's wrong: $a\le a^3$ holds for $a\ge 1$ but fails for $0<a<1$, where the cube is smaller. Correct approach: include fractional test values.

Mistake 2: Assuming a cube relation behaves symmetrically

Why it's wrong: cubing one side breaks the balance, so $1\le 2^3$ does not imply $2\le 1^3$. Correct approach: test the reverse pair explicitly. …