How to prove that $(\mathbb{Z}\backslash\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023\},\times)$ is a monoid?
Which numbers need to be verified to satisfy closedness for multiplication? Not sure where to start with verifying properties?
2026-03-26 19:01:48.1774551708
How to prove that (ℤ\{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023}, ×) is a monoid?
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Let $F$ be a subset of $\mathbb{Z}$. Then $\mathbb{Z} \setminus F$ is a monoid if $1 \notin F$ and $$ x \in \mathbb{Z} \setminus F \text{ and }y \in \mathbb{Z} \setminus F \implies xy \in \mathbb{Z} \setminus F $$ or equivalently $$ xy \in F \implies x \in F \text{ or } y \in F $$ It now suffices to verify that the set $F = \{-17, -7, -6, -3, -2, -1, 2, 3, 6, 7, 17, 2022, 2023\}$ satisfies this condition.