Q.Determine whether $*$ is a binary operation on the sets given below.

Step 1. Part (i): $a*b=a\cdot|b|$ on $\mathbb R$. For any $a,b\in\mathbb R$, $|b|$ is a well-defined non-negative real number, and $a\cdot|b|$ is the product of two real numbers -- always a single, well-defined real number. So $a*b$ is defined for every pair and always lands in $\mathbb R$.

Conclusion (i). $*$ is a binary operation on $\mathbb R$.

Step 2. Part (ii): $a*b=\min(a,b)$ on $A=\{1,2,3,4,5\}$. For any $a,b\in A$, $\min(a,b)$ is simply the smaller of the two numbers (or either one, if equal) -- and since both $a,b\in A$, the smaller of the two is also in $A$ (e.g. $\min(2,5)=2\in A$).

Conclusion (ii). $*$ is a binary operation on $A$.

Step 3. Part (iii): $(a*b)=a\sqrt b$ on $\mathbb R$. This requires $\sqrt b$ to be a real number, which fails whenever $b<0$ (e.g. $b=-4\Rightarrow\sqrt{-4}$ is not real). Take $a=1,b=-4\in\mathbb R\times\mathbb R$: $a*b=1\cdot\sqrt{-4}$ is not defined in $\mathbb R$ at all -- condition (i) of Definition 12.1 already fails.

Conclusion (iii). $*$ is not a binary operation on $\mathbb R$.

Check Definition 12.1's two conditions directly: defined everywhere, output stays in the set

• Forgetting that $\sqrt{b}$ needs $b\ge0$ to stay real
• Assuming closure without checking a negative or boundary value