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Suppose we have two sets $A$ and $B$ with $B \subseteq A$ does it follow that

$$ B \setminus \{ x \} \subseteq A \setminus \{ x \} $$

I cant find a counter example so i assume this is true in general?

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Suppose $y\in B\setminus\{x\}$. Then $y\in B$ and, since $B\subseteq A$, we have $y\in A$. Further, $y\in B\setminus\{x\}$ implies that $y\neq x$, and so $y\in A\setminus\{x\}$. Therefore, $B\setminus\{x\}\subseteq A\setminus\{x\}$.

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  • $\begingroup$ Note that $B\backslash\{x\}$ looks different from $B\setminus\{x\}.$ The latter has spacing appropriate to a binary operation symbol. I edited this answer accordingly. $\endgroup$ Jun 15 '17 at 0:42
  • $\begingroup$ That's good to know, thank you! $\endgroup$
    – florence
    Jun 15 '17 at 0:46
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If $b\in B\setminus \lbrace x\rbrace$, then $b\in B$ and $b\neq x$. Since $B$ is a subset of $A$, then we know $b\in A$. So $b\in A$ and $b\neq x$, by definition it follows that $b\in A\setminus \lbrace x\rbrace$.

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The statement $B\subseteq A$ means every member of $B$ is a member of $A$. The question is then whether every member of $B\smallsetminus\{x\}$ is a member of $A\smallsetminus\{x\}.$

Suppose $y$ is a member of $B\smallsetminus\{x\}.$ Then $y$ is a member of $B$ and $y\ne x.$ Since $y$ is a member of $B$ and every member of $B$ is a member of $A$, it follows that $y$ is a member of $A.$ And it is still true that $y\ne x.$

Thus every member of $B\smallsetminus\{x\}$ is a member of $A\smallsetminus\{x\}.$

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for any two sets $A,B$ the relation $B \subseteq A$ is equivalent to $B \cap A=B$. let $X$ be the complement of $\{x\}$ in $A \cup B \cup \{x\}$, so $A\setminus\{x\} = A \cap X$, and $B\setminus\{x\} = B \cap X$. we have $$ (A\setminus\{x\}) \cap (B\setminus\{x\}) = ( A \cap X) \cap (B \cap X) = (A \cap B) \cap X = B \cap X = B\setminus\{x\} $$ hence $$ B\setminus\{x\} \subseteq A\setminus\{x\} $$

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  • $\begingroup$ See my comment under the answer posted by "florence". I did the same correction on this answer as on that one. $\endgroup$ Jun 15 '17 at 1:27
  • $\begingroup$ thank you Michael. these little typesetting details are elusive! $\endgroup$ Jun 15 '17 at 11:18

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