Venn Diagrams
Venn diagrams illustrate set relationships, particularly negation and complement, while highlighting the law of excluded middle and the problem of grey areas.
THE IDEA
VENN DIAGRAMS
To think about negation, consider white people and non‑white people. We could have a Venn diagram of people, with a circle for white people; the space outside that circle is the region containing all the people who are not white. The complement contains everyone in that universe who does not belong to the chosen set. Racial and ethnic labels require care here: for example, Latin identity does not by itself determine whether someone identifies as white.
In general we can represent the statement “A is true” by drawing a circle labelled A. Then “A is not true” corresponds to everything outside that circle. In the language of sets and Venn diagrams this outside region is called the complement — the part that fits together with A perfectly, leaving no gaps. Here we come up against the law of the excluded middle: the complement takes up all the space outside the circle, and there is nothing in between that space and the circle itself. If we allowed a grey area it would look like a fuzzy boundary, but classical logic insists on a sharp dividing line. When the concepts are “white” and “not white”, that dividing circle runs through the middle ground; when we switch to “black” and “not black” the dividing circle sits at the edge, just where the deep black starts.
Once there is more than one set, we can see how logical statements connect with each other. Suppose we are thinking about people who, on average, earn less than the general population. Suppose black people suffer this disadvantage, as do women, but the people at the most grave disadvantage are those who are both black and women. We can depict this in a Venn diagram with a circle for black people and a circle for women. The region of overlap in the middle — where both things are true — is called the intersection, and in this case it consists of black women.
If instead we consider those who are at some kind of income disadvantage (not necessarily the most severe), then we need to include black people who are not women and women who are not black, as well as those who are both. This gives us the combined region from both circles, which is called the union of the two sets.
Intersections represent “and”; unions represent inclusive “or”. Complements represent negation within the chosen universe.
When to use it
Teaching basic set theory and logical negation in mathematics classrooms.
Analyzing categorical data with explicit membership rules, such as respondents who answered yes to a particular question and those who did not.
Visualizing the logical relationship between a statement and its negation in philosophy or critical thinking exercises.
Identifying gaps in category definitions when designing surveys or classification systems.
Comparing customer segments that have multiple overlap properties for targeted marketing.
Solving probability problems where events can occur simultaneously (e.g., drawing a card that is both red and a face card).
Analyzing survey data to find groups of respondents who satisfy multiple criteria.
Teaching logical reasoning in educational settings.
FROM THEORY TO EVERYDAY LIFE
In practice
White vs. Non-White
Within a defined population, draw a circle for people classified as white under the chosen definition. Its complement contains everyone outside that circle. Other racial and ethnic labels should not be assumed to be mutually exclusive.
A Statement Truth
For a statement 'A is true', draw a circle for A. The region where A is not true is the entire area outside the circle, illustrating the law of excluded middle.
Income Disadvantage Intersection
The text uses the example of black women experiencing the most severe income disadvantage. This is the intersection of the sets 'black people' and 'women', where both conditions are true.
Union of Disadvantages
When considering anyone at some kind of income disadvantage, we include all individuals in either set (black people or women), which is the union of the two sets.
A MOMENT TO REFLECT
Put it to work.
How does the law of excluded middle apply when categorizing ambiguous cases (e.g., biracial individuals)?
When a category has ambiguous boundaries, how should you clarify its definition before using a binary set diagram?
How would you represent a concept with fuzzy boundaries using a Venn diagram? What modifications would be needed?
How can you use a Venn diagram to analyze overlapping characteristics in a group of people or items?
What is the difference between 'and' (intersection) and 'or' (union) in terms of set membership?
Can you think of a real-world situation where understanding the intersection of two conditions is more important than the union?
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)