TITLE: Max length of meaningful combinations of union and intersection of three sets.
QUESTION [5 upvotes]: Given three sets: $X, Y, Z$ and the two set operations: union and intersection. 
What is the maximum length of a 'formula' which is not reducible to a shorter formula. 
Eg. the formula $(X \cap Y) \cup X$ can be reduced to the formula $X$. 
However the formula $Y \cap(X \cup Z)$ is not reducible in length. 
I've been thinking about this problem for a while now, but can't seem to get past a length of 3. (the length being the amount of set symbols).

REPLY [3 votes]: All possible combinations (I think):
$X$
$Y$
$Z$
$X \cup Y$
$Y \cup Z$
$X \cup Z$
$X \cap Y$
$Y \cap Z$
$X \cap Z$
$(X \cap Y) \cup Z$
$(Y \cap Z) \cup X$
$(Z \cap X) \cup Y$
$(X \cup Y) \cap Z$
$(Z \cup X) \cap Y$
$(Y \cup Z) \cap X$
$X \cap Y \cap Z$
$X \cup Y \cup Z$
$\{(X \cup Z) \cap Y\} \cup (X \cap Z)$
Venn says: