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

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

$
A\subseteq B$
: given

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

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

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

$
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
$
A\subseteq B$
: given

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

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

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

$
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.
$\begin{gathered}
q \in R\left[ A \right]\; \Rightarrow \;\left( {\exists p \in A} \right)\left[ {\left( {p,q} \right) \in R} \right] \hfill \\
A \subseteq B\; \Rightarrow \;p \in B\; \Rightarrow \;\left( {q \in R\left[ B \right]} \right) \hfill \\
R\left[ A \right] \subseteq R\left[ B \right] \hfill \\
\end{gathered}$

3. Originally Posted by Plato
$

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 $(p,q)\in R \text{ and } p\in B$ so by definition $q\in R[ B ]$.