Inclusive vs. Exclusive Definitions

Date: 01/24/2002 at 15:44:09
From: Logan Rhyne
Subject: Quadrilaterals

Dr. Math,

My geometry teacher says that a square is not also a rhombus, a 
rectangle, and a parallelogram. I cannot convince him that this is not 
so. Please help!

Date: 01/24/2002 at 16:37:55
From: Doctor Peterson
Subject: Re: Quadrilaterals

Hi, Logan.

Both "inclusive" and "exclusive" definitions are used for such things; 
you and I agree that the inclusive definition (rectangles include 
squares) is more useful than the exclusive definition (rectangles must 
have unequal length and width) that is often taught to children. We 
can either convince your teacher of this judgment, using arguments 
like those here:

    Inclusive and Exclusive Definitions
    http://mathforum.org/dr.math/problems/hawes.04.05.01.html   

or we can just show that both definitions are valid, so that you are 
at least not wrong. Try a dictionary definition like this, from 
Merriam-Webster (m-w.com):

    Main Entry: rect.an.gle 
    Function: noun
    Etymology: Medieval Latin rectangulus having a right angle, from
    Latin rectus right + angulus angle -- more at RIGHT, ANGLE
    Date: 1571
      a parallelogram all of whose angles are right angles; especially
      one with adjacent sides of unequal length 

This says that the word can be taken in general of any right-angled 
parallelogram, or more specifically of one that is not a square. 
Similarly,

    Main Entry: rhom.bus 
    Function: noun
    Inflected Form(s): plural rhom.bus.es or rhom.bi  /-"bI, -"bE/
    Etymology: Latin, from Greek rhombos piece of wood whirled on a
    string, lozenge, from rhembein to whirl
    Date: circa 1567
      a parallelogram with four equal sides and sometimes one with no
      right angles 

More clearly this time, the word only _sometimes_ excludes right 
angles.

But notice that both of these are defined as parallelograms! There's 
no doubt that that is defined inclusively.

Finally, check out "square":

    Main Entry: square 
    Function: noun
    Etymology: Middle English, from Middle French esquarre, from
    (assumed) Vulgar Latin exquadra, from exquadrare to square, from
    Latin ex- + quadrare to square -- more at QUADRATE
    Date: 13th century
    ...
    2 : a rectangle with all four sides equal
    ...

So a square _is_ a rectangle.

Now, dictionaries don't always get math terms right, but they do 
carefully research how words are actually used. The fact that they 
give first the definition we prefer, which is probably more popular 
among mathematicians than in the general public, seems to support our 
contention. Now let's look at a scientific dictionary and check their 
view:

    Harcourt Academic Press Dictionary of Science and Technology
    http://www.harcourt.com/dictionary/def/9/7/2/5/9725100.html   

    square   Mathematics.  1. a quadrilateral having all four sides 
    and all four angles equal; equivalently, a rectangle with equal 
    sides or a rhombus with a right angle.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/