Illustrated · 1 min
You may have heard the following classic debate in naming quadrilaterals: should a square be considered a rectangle? Should a parallelogram be considered a trapezoid?
In other words, should we use inclusive definitions (a square is a rectangle, a parallelogram is a trapezoid, etc.) or exclusive definitions (a square is not a rectangle, a parallelogram is not a trapezoid, etc.)? There is no objectively right or wrong answer to these questions. Here are some simple arguments:
- The area formula for a rectangle (base times height) works perfectly well for a square. So maybe a square should be considered a rectangle.
- A square has more lines of symmetry and more rotational symmetry than a rectangle with two different side lengths. So maybe a square shouldn't be considered a rectangle?
However, one advantage of the inclusive definitions is the following nice "quadrilateral hierarchy" or "family tree", which goes from "general" (quadrilateral with no requirements) to "specific" (squares, required to have four congruent sides and four congruent right angles).
Experience reveals another advantage: as pointed out by the famous mathematician John Conway, natural inclusive definitions tend to simplify statements. For example, you may have seen the following fact before:
A parallelogram with congruent diagonals is a rectangle.
The point is: as stated, this fact is only correct if we use the inclusive definition for rectangles. If we used the exclusive definition for rectangles, the statement would be false as written, and would need to be corrected to the following:
A parallelogram with congruent diagonals is a rectangle or a square.
At the end of the day, it's a matter of taste whether to use inclusive or exclusive definitions. However, the reasons above hopefully illustrate the conceptual advantages of inclusive definitions.