A. Examples of operations

1 The operation $a \ast b = \sqrt{|ab|}$ is not an operation on $\mathbb{Q}$, because there are pairs of rational numbers whose square of absolute value their product is not in $\mathbb{A}$. For example, if $a = 1$ and $b = 2$, $a$ and $b$ are in $\mathbb{Q}$, but $a \ast b$ $=$ $1 \ast 2$ $=$ $\sqrt{|1(2)|}$ $=$ $\sqrt{2}$ is not.

2 The operation $a \ast b = a \ln b$ is not operation on the set $\{ x \in \mathbb{R} \ : \ x>0 \}$, because even if $a = 1$ and $b = 2^{-1}$ are in the given set, $a \ast b$ $=$ $1 \ast 2^{-1}$ $=$ $1 \ln 2^{-1}$ $<$ $0$ is not.

3 Given the quadratic equation $x^2$ $-$ $a^2b^2$ $=$ $0$, it can be seen that

$$ \phantom{,} \ x^2 - (ab)^2 = (x - ab)(x + ab) = 0 \ , $$

so when solving the equation for $x$, the roots are either $ab$ or $-(ab)$, which means the for given $x$ in $\mathbb{R}$, the operation

$$ a \ast b = x^2 - a^2b^2 $$

is not operation on $\mathbb{R}$, because $a \ast b$ as a root is not uniquely defined.

4 Subtraction is operation on $\mathbb{Z}$, because it is defined for every ordered pair in $\mathbb{Z}$, each subtraciton $(a - b)$ in uniquely defined and exists only in $\mathbb{Z}$ — that is, not outside the set.

5 Subtraction is not an operation on the set $\{n \in \mathbb{Z} \ : \ n \geq 0 \}$, because for any $a$ strictly smaller than any $b$ in the set, subtraction, that is $a \ast b$ $=$ $a - b$ is not in the set.

6 The operation $a \ast b$ $=$ $|a - b|$ is an operation on the set $\{n \in \mathbb{Z} \ : \ n \geq 0 \}$, since by definition of $| \ \cdot \ |$, all values are greater or equal and never less than $0$, that is, the absolute value of subtraction of any integer is always non-negative. Moreover, each element $|a - b|$ is defined and unique for all $a$ and $b$ and all absolute values of subtraction between the elemens exists in the given set.