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I'm having a hard time trying to demonstrate the definition of a subset, which is :

$S \subseteq T \Leftrightarrow (S \subset T) \vee (S = T)$

I always end up by expending the definitions of $\subset$ and $=$ in Boolean algebra, resulting into this complicated situation.

Any way to solve it by staying in the Set theory? Or in Boolean algebra?

Andrés E. Caicedo
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lolgab123
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1 Answers1

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You're on the right way!

Your definition of $S \subset T$ as $\forall x (x \in S \rightarrow x \in T) \land \exists x (x \in T \land \neg x \in S)$ should work fine.

For $S = T$ you can use $\forall x (x \in S \leftrightarrow x \in T)$

And for $S \subseteq T$ you can use $\forall x (x \in S \rightarrow x \in T)$

Using these definitions, you should be able to prove $S \subseteq T \leftrightarrow (S \subset T \lor S =T)$ just fine.

Bram28
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  • Actually I have made a little mistake. The thing is that I start by proving the $\Rightarrow$, supposing that $S \subseteq T$ is true, then for the right-side equation, the simplification results in $(\exists e \mid e \in S \wedge e \notin T) \vee (\forall e \mid e \in T \Rightarrow e \in S)$, from which I'm totally unable to continue... – lolgab123 Oct 03 '17 at 02:14
  • @lolgab123 What you have there is not the simplification of the right-side. To peove the right side from the left side, consider that either $\exists x (x \in T \land \neg x \in S)$ or $\neg \exists x (x \in T \land \neg x \in S)$ – Bram28 Oct 03 '17 at 02:39