How do Euler diagrams show the relationships between categories?

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Marble bust of Aristotle with a beard and draped robe, showing detailed facial features and wavy hair, set against a dark background. — Applied logic Aristotle, marble portrait bust, Roman copy (2nd century bce) of a Greek original (c. 325 bce); in the Museo Nazionale Romano, Rome.

An Euler diagram is a graphic tool used to represent the logical relationships between categories or classes. It utilizes circles to visualize how different groups interact through three primary operations:1

Inclusion: One circle contained entirely within another indicates that all members of one category are members of the other.
Exclusion: Two circles that do not overlap indicate that the categories share no common members.
Intersection: Overlapping circles indicate that the categories share some, but not necessarily all, members.

By using these spatial arrangements, Euler diagrams provide a clear method for mapping the terms of propositions and understanding the logical structure of categorical relationships.1