TITLE: Trivial question about set difference: does $B\subseteq A$ imply $B\setminus\{x\}\subseteq A\setminus\{x\}$?
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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
$$
 B\setminus\{x\} \subseteq  A\setminus\{x\} 
$$