diff options
Diffstat (limited to 'library/set.tex')
| -rw-r--r-- | library/set.tex | 14 |
1 files changed, 11 insertions, 3 deletions
diff --git a/library/set.tex b/library/set.tex index 69b9526..acbc71c 100644 --- a/library/set.tex +++ b/library/set.tex @@ -385,6 +385,11 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio Follows by set extensionality. \end{proof} +\begin{proposition}\label{union_subsets_is_subset} + Suppose $A\subseteq C$ and $B\subseteq C$. + Then $A\union B\subseteq C$. +\end{proposition} + \begin{proposition}\label{subseteq_union_iff} $A\union B\subseteq C$ iff $A\subseteq C$ and $B\subseteq C$. \end{proposition} @@ -626,7 +631,7 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio \begin{proposition}[Binary intersection over binary union]% \label{inter_distrib_union} - $x\inter (y\union z) = (x\inter y)\union (x\inter z)$. + $A\inter (B\union C) = (A\inter B)\union (A\inter C)$. \end{proposition} \begin{proof} Follows by set extensionality. @@ -643,10 +648,13 @@ The $\operatorname{\textsf{cons}}$ operation is determined by the following axio % Halmos, Naive Set Theory, page 16 \begin{proposition}\label{union_inter_assoc_intro} Suppose $C\subseteq A$. - Then $(A\inter B)\union C = A\inter (B\union C)$. + Then $A\inter (B\union C) = (A\inter B)\union C$. \end{proposition} \begin{proof} - Follows by set extensionality. + \begin{align*} + A\inter (B\union C) &= (A\inter B)\union (A\inter C) \explanation{by \cref{inter_distrib_union}}\\ + &= (A\inter B)\union C \explanation{by \cref{inter_absorb_supseteq_left}} + \end{align*} \end{proof} % Halmos, Naive Set Theory, page 16 |