# Proof by contradiction: $A \subseteq A$

I have to proof by contradiction that $A \subseteq A$; if $A \nsubseteq A$ then $\exists x \in A ( x \notin A )$ but $\exists x \in A ( x \notin A )$ is contradiction (in fact: $\exists x ( x \in A \wedge x \notin A )$) therefore $A \subseteq A$ is true... The process is correct? Thank you all in advance

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Yes, this is fine. –  Brian M. Scott Mar 14 '13 at 15:40
Yes its correct –  kalpeshmpopat Mar 14 '13 at 15:41
Thank you soo much!!! –  mle Mar 14 '13 at 15:48

Your proof is correct, the way I would write it is as follows:

Suppose there exists a set $A$ such that $A \nsubseteq A$. Then there exists $x \in A$ such that $x \notin A$. But $x \in A$, contradiction. Hence $A \subseteq A$.

The extra formalism isn't necessary unless you are trying to get a better understanding of using the quantifiers, which in this case you have used correctly.

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It looks fine but it is better you clarify both $A$ as $$A_1\subseteq A_2$$ where $A_1=A_2=A$.

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I don't think I agree. This just encourages you to believe that things are different when they are actually the same. The contradiction is clearer if you write "$x\in A$ and $x\notin A$" than if you write "$x\in A_1$, $x\notin A_2$ and $A_1=A_2$". But maybe there are other reasons to prefer this notation. –  Matt Pressland Mar 14 '13 at 15:42
@MattPressland: I know it might make some points you noted but if I wanted to prove like the OP did, I would prefered student find out what $A$ I mean when I write $x\in A$, the first $A$ or the second one. However, they both are the same. –  Babak S. Mar 14 '13 at 15:45
That is also reasonable. My feeling to a degree though is that the contradiction arises from the fact that when you write $x\in A$, you can't tell "which $A$" you mean - you have to mean both - so you can't possibly also have $x\notin A$. –  Matt Pressland Mar 14 '13 at 15:48
@MattPressland: Sometimes, I feel need a blackboard here to say what I mean. –  Babak S. Mar 14 '13 at 15:54
It's fine, this is just a semantic argument anyway. Different people will find different notation clearer. –  Matt Pressland Mar 14 '13 at 15:57