# $x\notin X\setminus\{y\} \iff x\notin X$ or $x=y$

Is the following deduction correct?

\begin{align} x\notin X\setminus\{y\}\quad & \iff\quad x\notin X \cap \overline{\{y\}}\\ \\ & \iff\quad x \in \overline{X\cap \overline{\{y\}}}\\ \\ & \iff\quad x \in \overline X \cup \{y\}\\ \\ & \iff\quad x \notin X\text{ or } x =y\end{align}

• Yup. ${}{}{}{}$ – Clive Newstead Sep 18 '13 at 13:22
• I would strongly advise against using $\overline{X}$ to denote the complement - it denotes the closure - but apart from that, I see nothing to criticize. – Daniel Fischer Sep 18 '13 at 13:22
• What about the obvious $x\in X\setminus\{y\}\iff x\in X \land x\neq y$? – Michael Hoppe Sep 18 '13 at 13:50

So I'd suggest adding more words, if only to add justification. For e.g., you use DeMorgan's when going from the third to fourth line: $$\quad x \in \overline{X\cap \overline{\{y\}}} \iff x \in \overline X \cup \{y\}\tag{DeMorgan's}$$ Say so!
What you did is fine but I'd negate both sides and then it's obvious by definition: $$x\in X\setminus\{y\} \iff x\in X \text{ and } x\ne y$$