Does the size of a Venn diagram indicate the cardinality of the set?

Question

Since a Venn diagram is a schematic representation, does the size of the Venn diagram have any meaning?

Answer 1

Usually, no, because just reproducing faithfully the intersections on a $2$-d sheet of paper is enough of a technical difficulty without bothering with estimates of the areas.

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The usual way to encode cardinality effectively in a Venn diagram is by labelling each region with its own.

Answer 2

No. The whole point of Venn diagrams is simply to visualize all possibilities as far as objects belonging or not belonging to some category or set of objects.

Indeed, even when a set $A$ is a subset of a set $B$, you don't draw set $A$ completely within set $B$; that's something an Euler Diagram would do, but not a Venn diagram. If in a Venn diagram you want to indicate that $A$ is a subset of $B$, you can 'black out' the region inside $A$ and outside $B$, but you still show that region. So, the best you can do with Venn diagrams is to show that there are or are not objects in the possible regions, but there is no indication of relative cardinality, let alone absolute soze.