Q.Let us define a relation $R$ in $\mathbb{R}$ as $aRb$ if $a \geq b$. Then $R$ is
(A) an equivalence relation
(B) reflexive, transitive but not symmetric
(C) symmetric, transitive but not reflexive
(D) neither transitive nor reflexive but symmetric

The key idea is that $a \geq b$ defines a reflexive and transitive relation, but it is not symmetric.

Step 1 — Reflexive: For any $a \in \mathbb{R}$, $a \geq a$ is always true. So $R$ is reflexive.

Step 2 — Symmetric? If $a \geq b$, does it imply $b \geq a$? Only when $a = b$. For example, $3 \geq 2$ is true, but $2 \geq 3$ is false. So $R$ is not symmetric. …

The core of this problem is checking three properties — reflexivity, symmetry, and transitivity — against the definition $aRb \iff a \geq b$. Each property tests a different logical condition, and the key is to apply them to real numbers without overcomplicating.

Let’s go step by step.

1. Reflexivity: A relation $R$ on a set is reflexive if every element is related to itself. For $aRb$ to hold when $a = b$, we need $a \geq a$. Since any real number is equal to itself, $a \geq a$ is always true. So $R$ is reflexive.

2. Symmetry: A relation is symmetric if whenever $aRb$ holds, then $bRa$ must also hold. Here, $aRb$ means $a \geq b$. Does $a \geq b$ always imply $b \geq a$? Only if $a = b$. For a counterexample, take $a = 5$ and $b = 3$: $5 \geq 3$ is true, but $3 \geq 5$ is false. So symmetry fails.

Method: Finding the identity element of a binary operation

Use this when asked for the identity element $e$ of an operation $a*b$ on a set.

Steps

Step 1: Write the defining equation for the identity.

$e$ is the identity iff $a*e=a$ (and $e*a=a$) for EVERY $a$ in the set. Do not guess $e=0$ or $e=1$ out of habit — solve for it.

Step 2: Substitute the rule and solve for $e$.

Replace $a*e$ using the definition and set it equal to $a$. For $a*b=\dfrac{ab}{2}$ this gives $\dfrac{ae}{2}=a$; since $a\neq 0$, divide by $a$ to get $\dfrac{e}{2}=1$, so $e=2$.

Step 3: Check the other side (unless you have already shown commutativity). …

Common Mistakes

Mistake 1: Guessing $e=1$ because $1$ is the identity for ordinary multiplication.

Why it's wrong: the operation is $a*b=\dfrac{ab}{2}$, not $ab$; $a*1=\dfrac{a}{2}\neq a$. Correct approach: solve $a*e=a$ from the actual rule, giving $\dfrac{ae}{2}=a\Rightarrow e=2$.

Mistake 2: Checking only one side of the identity.

Why it's wrong: an identity must satisfy both $a*e=a$ and $e*a=a$. Correct approach: verify $e*a=\dfrac{2a}{2}=a$ as well (here it works because $*$ is commutative). …