# The usefulness of classification

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

• polygon
• convex polygon
• trapezoid
• isosceles trapezoid
• 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.