# Proof by contradiction: $\emptyset \subseteq A$

I have to proof by contradiction that: let $A$ a set and $\emptyset$ the empty set, then $\emptyset \subseteq A$; if $\emptyset \nsubseteq A$ then $\exists x \in \emptyset ( x \notin A )$ but for hypothesis "let $\emptyset$ the empty set, then $\nexists x \in \emptyset$", so I have a contradiction and therefore $\emptyset \subseteq A$ is true! Is it correct? Thank you all in advance

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Yes, this also is just fine. – Brian M. Scott Mar 14 '13 at 16:15
Ya thats correct answer – kalpeshmpopat Mar 14 '13 at 16:23
@BrianM.Scott thank you! – mle Mar 14 '13 at 18:04
@kalpeshmpopat thank you! – mle Mar 14 '13 at 18:05
You’re welcome. – Brian M. Scott Mar 14 '13 at 18:05

## 2 Answers

Nitpick (very slight alteration to follow):

Let $A$ [be] a set and $\emptyset$ the empty set. Then $\emptyset \subseteq A$.

Proof:
[Let $A$ be a set and $\emptyset$ the empty set. Suppose also, for the sake of contradiction, that] $\; \emptyset \nsubseteq A$.
Then $\exists x \in \emptyset,$ [such that] $( x \notin A )$.
But by hypothesis, $\emptyset$ is the empty set, [thus by the definition of the empty set], $\lnot\exists x \in \emptyset$.
So [we] have [reached] a contradiction, and it must therefore follow that] $\;\;\emptyset \subseteq A,$ [as desired].

(Note: here $\lnot\exists \equiv \nexists$)

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for tomorrow morning – Babak S. Mar 14 '13 at 20:11
@Oleg: you had all the right ideas and your proof does indeed work. My suggestions are not implying you were wrong: I am simply suggesting how to "tighten up" your proof, and how you can make it a tad more "formal" and precise. – amWhy Mar 14 '13 at 20:37
@amWhy thank you for idea!! – mle Mar 14 '13 at 21:02

Yes, that is a correct proof of your statement by contradiction.

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