Under which of the following operations are the integers closed? Select all correct answers.
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Which of the following rules are operations on the indicated set? ( $\mathbb{Z}$ designates the set of the integers, $\mathbb{Q}$ the rational numbers, and $\mathbb{R}$ the real numbers.) For each rule which is not an operation, explain why it is not. Example $a * b=\frac{a+b}{a b}$, on the set $\mathbb{Z}$. SoLUTION This is not an operation on $\mathbb{Z}$. There are integers $a$ and $b$ such that $(a+b) / a b$ is not an integer. (For example, $$ \frac{2+3}{2 \cdot 3}=\frac{5}{6} $$ is not an integer.) Thus, $\mathbb{Z}$ is not closed under $*$ Subtraction, on the set $\mathbb{Z}$.
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Which of the following rules are operations on the indicated set? ( $\mathbb{Z}$ designates the set of the integers, $\mathbb{Q}$ the rational numbers, and $\mathbb{R}$ the real numbers.) For each rule which is not an operation, explain why it is not. Example $a * b=\frac{a+b}{a b}$, on the set $\mathbb{Z}$. SoLUTION This is not an operation on $\mathbb{Z}$. There are integers $a$ and $b$ such that $(a+b) / a b$ is not an integer. (For example, $$ \frac{2+3}{2 \cdot 3}=\frac{5}{6} $$ is not an integer.) Thus, $\mathbb{Z}$ is not closed under $*$ Subtraction, on the set $\{n \in \mathbb{Z}: n \geqslant 0\}$
Which of the following rules are operations on the indicated set? ( $\mathbb{Z}$ designates the set of the integers, $\mathbb{Q}$ the rational numbers, and $\mathbb{R}$ the real numbers.) For each rule which is not an operation, explain why it is not. Example $a * b=\frac{a+b}{a b}$, on the set $\mathbb{Z}$. SoLUTION This is not an operation on $\mathbb{Z}$. There are integers $a$ and $b$ such that $(a+b) / a b$ is not an integer. (For example, $$ \frac{2+3}{2 \cdot 3}=\frac{5}{6} $$ is not an integer.) Thus, $\mathbb{Z}$ is not closed under $*$ $a: b=|a-b|$, on the set $\{n \in \mathbb{Z}: n \geqslant 0\}$
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