Q.Examine whether the operation $*$ defined on $R$ by $a * b = ab + 1$ is

The key idea is to check whether $*$ is a binary operation (closed on $\mathbb{R}$) and then test associativity.

Step 1: Binary operation check.

For any $a, b \in \mathbb{R}$, $ab + 1$ is a real number. So $*$ maps $\mathbb{R} \times \mathbb{R}$ to $\mathbb{R}$ — it is a binary operation.

Step 2: Associativity test.

We need to check if $(a * b) * c = a * (b * c)$ for all $a, b, c \in \mathbb{R}$.

Compute left side:

$(a * b) * c = (ab + 1) * c = (ab + 1)c + 1 = abc + c + 1$.

Compute right side:

$a * (b * c) = a * (bc + 1) = a(bc + 1) + 1 = abc + a + 1$. …

Why this approach works

Before we check anything, we need to be clear on two definitions.

A binary operation on a set $S$ is simply a rule that takes any two elements of $S$ and produces another element of $S$. That’s it — the only requirement is closure: the result must stay inside $S$.

Associativity is a separate property. An operation $*$ is associative if the grouping of operations doesn’t matter:

$(a * b) * c = a * (b * c)$ for all $a, b, c$ in the set.

The natural strategy is: first verify closure (is it a binary operation?), then test associativity by computing both sides of the associative law and comparing them.

Step-by-step solution

1. Check if $*$ is a binary operation on $\mathbb{R}$

   For any $a, b \in \mathbb{R}$, the expression $a * b = ab + 1$ involves multiplication and addition of real numbers. Both operations are closed in $\mathbb{R}$ — the product $ab$ is a real number, and adding $1$ keeps it real.

   So $a * b \in \mathbb{R}$ for every $a, b \in \mathbb{R}$.

   Hence $*$ is a binary operation on $\mathbb{R}$.

2. Test associativity: compute $(a * b) * c$

   First apply the definition to $a$ and $b$:

Now treat this result as the left operand with $c$:

Simplify:

3. Now compute $a * (b * c)$ First find $b * c$:

Then combine with $a$:

Simplify:

4. Compare the two results We have:

For associativity to hold, these must be equal for all $a, b, c \in \mathbb{R}$. …

Method: Checking Closure and Associativity of a Binary Operation

Use this when a rule $a*b$ is given and you must decide (i) whether it is a valid binary operation and (ii) whether it is associative.

Steps

Step 1: Test closure (is it binary?)

Confirm that $a*b$ always lands back in the set. If $a,b\in\mathbb{R}$ and $a*b=ab+1$ is again real, the operation is binary on $\mathbb{R}$.

Step 2: Set up both groupings for associativity

Associativity means $(a*b)*c=a*(b*c)$ for all $a,b,c$. Compute each side separately, treating one result as an operand in the next step.

Step 3: Expand carefully …

Common Mistakes

Mistake 1: Assuming $ab+1$ is associative because $\times$ and $+$ are

Why it's wrong: combining associative operations with an extra constant can break associativity, since the $+1$ attaches to a different operand depending on the grouping. Correct approach: expand both groupings and compare, never assume.

Mistake 2: Testing associativity with a single triple that happens to work …