At the Lapinou family's latest meal, 80% of the family members ate carrots, 45% ate salad, and 30% ate both. What proportion ate carrots or salad—that is, at least one of the two foods?
A quick sketch is worth a thousand words. Just look at the image below, where C is the set of rabbits that ate carrots and S the set of those that ate salad. If we add the two respective proportions, the picture shows that we have counted the rabbits in the intersection of the two sets once too often—that is, those that ate both carrots and salad. The proportion we seek is therefore 80% + 45% – 30% = 95%. If pC and pS denote the proportions of the sets C and S, we can even derive a more general formula:
pC∪S=pC+pS−pC∩S.p_{C\cup S}=p_{C}+p_{S}-p_{C\cap S}.
Leonhard Euler was already using diagrams of this kind in the 18th century, though in a slightly different context: to illustrate logical expressions.
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