# The usefulness of classification

Why is understanding the relationships between sets useful? Consider the following shape:

What do we know about this particular shape?

- polygon
- convex polygon
- quadrilateral
- convex quadrilateral
- trapezoid
- isosceles trapezoid
- cyclic quadrilateral
- tiles the plane
- rep-tile

Notice that some of these properties are redundant. For example, if I know something is a trapezoid, I also know it's a quadrilateral. Here are the relevant set relations, for this particular list of properties.

- Every convex polygon is a polygon, by definition.
- In other words, the set of convex polygons, is a subset of the set of polygons.
- Every quadrilateral is a polygon, by definition.
- In other words, the set of quadrilaterals, is a subset of the set of polygons.
- Every convex quadrilateral is a convex polygon, and a quadrilateral, by definition.
- In other words, the set of convex quadrilaterals, is the intersection of the set of convex polygons, and the set of quadrilaterals.
- Every trapezoid is a quadrilateral, by definition.
- In other words, the set of trapezoids, is a subset of the set of quadrilaterals.
- A trapezoid is cyclic if and only if it's isosceles.
- In other words, the set of cyclic trapezoids is equivalent to the set of isosceles trapezoids.
- Every convex quadrilateral tiles the plane.
- In other words, the set of convex quadrilaterals, is a subset of the set of monohedral prototiles.
- Every rep-tile tiles the plane.
- In other words, the set of rep-tiles, is a subset of the set of monohedral prototiles.

Notice that some set relations are established by definition, while others are established by proof. Now we have a list of set relations, which we can apply to our initial list of properties. We find our initial list boils down, to just

- isosceles trapezoid
- rep-tile

The beauty of knowing the essential properties of this shape, is that now we can ask: "Are there any other isosceles trapezoid rep-tiles, or is this the only one? I will leave that for you to discover, in case you don't already know the answer.