Thread: [SOLVED] How Much of a Proof is this

1. [SOLVED] How Much of a Proof is this

$\displaystyle A\subseteq B$ : given

$\displaystyle R[A]=\{y\mid(\exists x\in A)(x,y)\in R\}$
$\displaystyle R[ B ]=\{y\mid(\exists x\in B)(x,y)\in R\}$
$\displaystyle (\forall x\in A)\,\, x\in B$
$\displaystyle (\forall y\in R[A])\,\, y\in\{y\mid(\exists x\in B)\,(x,y)\in R\}$

$\displaystyle R[A]\subseteq R[ B ]$

I have to prove the last line in that list; does this constitute a proof?

Thanks, I am unused to both set theory and proofs, which makes this especially hard.

2. Originally Posted by billa
$\displaystyle A\subseteq B$ : given

$\displaystyle R[A]=\{y\mid(\exists x\in A)(x,y)\in R\}$
$\displaystyle R[ B ]=\{y\mid(\exists x\in B)(x,y)\in R\}$
$\displaystyle (\forall x\in A)\,\, x\in B$
$\displaystyle (\forall y\in R[A])\,\, y\in\{y\mid(\exists x\in B)\,(x,y)\in R\}$

$\displaystyle R[A]\subseteq R[ B ]$

I have to proof the last line in that list; does this constitute a proof?

Thanks, I am unused to both set theory and proofs, which makes this especially hard.
$\displaystyle \begin{gathered} q \in R\left[ A \right]\; \Rightarrow \;\left( {\exists p \in A} \right)\left[ {\left( {p,q} \right) \in R} \right] \hfill \\ A \subseteq B\; \Rightarrow \;p \in B\; \Rightarrow \;\left( {q \in R\left[ B \right]} \right) \hfill \\ R\left[ A \right] \subseteq R\left[ B \right] \hfill \\ \end{gathered}$

3. Originally Posted by Plato
$\displaystyle A \subseteq B\; \Rightarrow \;p \in B\; \Rightarrow \;\left( {q \in R\left[ B \right]} \right) \hfill \\$
excuse me, but how did you get that?

4. Originally Posted by doresa
excuse me, but how did you get that?
Well it is very clear.
We know that $\displaystyle (p,q)\in R \text{ and } p\in B$ so by definition $\displaystyle q\in R[ B ]$.