Vtyjkyuk

I've just started databases and have exercise to proof that projection$(pi)$ is distributive over set union$(bigcup)$. But I suck in proofs and don't really know how to proof that:

$pi_alpha( R bigcup S) = pi_alpha(R) bigcup pi_alpha(S) $

It's just to obvious and I don't know how to proof it.

References:

Definition of $bigcup: A bigcup B = x : x in A vee x in B $

Property of $pi $ $pi_alpha(pi_beta (R)) = pi_alpha(R) \where alpha subseteq beta $

I'm not sure about the notation, but I'll make an effort...

Say, tuples of $R$ have properties $R_1timescdotstimes R_n$ and tuples of $S$ have properties $S_1timescdotstimes S_m$. Without loss of generality, we assume $nleq m$.

Let $xinpi_a(Rcup S)$, where $asubseteq1,ldots,n$. Then
$$forall iin a, x_iinpi_i(Rcup S)text, where $x=prod_iin ax_i$Leftrightarrow\
forall iin a, x_iin R_icup S_iLeftrightarrow\
forall iin a, x_iin R_itext or x_iin S_iLeftrightarrow\
forall iin a, x_iin R_itext or forall iin a, x_iin S_iLeftrightarrow\
xinpi_a(R)text or xinpi_a(S)Leftrightarrow\
xinpi_a(R)cuppi_a(S)$$

Hence $pi_a(Rcup S)=pi_a(R)cuppi_a(S)$.