Inclusive and Exclusive Definitions

Date: 04/05/2001 at 17:46:14
From: hawesrd
Subject: Squares and Rectangles

Are squares rectangles? Are rectangles squares?

Thanks.

Date: 04/06/2001 at 06:08:25
From: Doctor Floor
Subject: Re: Squares and Rectangles

Hi,

Thanks for writing.

Rectangles are not always squares, because squares need four sides of 
equal lengths. The question whether the other way around is true 
depends on the type of definition you are using. There are often 
disputes about definitions.

In general there are two types of definition for geometric shapes:

INCLUSIVE DEFINITIONS: In the case of rectangles and squares this 
means that a square is seen as a special case of a rectangle:

 * A rectangle is a quadrilateral with four right angles.

 * A square is a quadrilateral with four right angles and four equal    
   sides.

EXCLUSIVE DEFINITIONS: In the case of rectangles and squares this 
means that a square is NOT considered a rectangle:

 * A rectangle is a quadrilateral with four right angles but not four 
   equal sides.

 * A square is a quadrilateral with four right angles and four equal 
   sides.

I prefer inclusive definitions, because they include the basic 
mathematical concept of 'generalization': one item (rectangle) is more 
general than the other item (square). But others say that exclusive 
definitions are very useful for special cases.

You can read a discussion about this in the geometry pre-college 
newsgroup at:

  Trapezoid definition
  http://mathforum.org/kb/message.jspa?messageID=1080683   

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/   

Date: 04/03/2001 at 20:40:02
From: Michelle
Subject: Isosceles triangles?

I am in the 10th grade and I need to know whether an isosceles 
triangle has exactly or only two sides congruent. Is an equilateral 
triangle considered isosceles as well? I've looked at over 30 sites 
but I never get a full answer on whether there are exactly, only, or 
at least 2 sides congruent - or if I do, there is no explanation or 
reasoning to prove the statement.

Could you please help me?

Thanks so much,
Michelle

Date: 04/04/2001 at 01:39:07
From: Doctor Schwa
Subject: Re: Isosceles triangles?

Hi Michelle,

The question of whether an isosceles triangle has to have at least two 
sides congruent versus exactly two sides congruent isn't something you 
can prove: it's a question of definition. What does the term 
"isosceles triangle" mean?

Definitions are something that we can choose arbitrarily, and books can 
have different definitions.

However, some definitions are more useful than others, more 
convenient, easier to use... and in this case, one of those two 
definition choices is much better than the other. The inclusive 
definition is almost always better, as it is in this case. The 
inclusive definition, where isosceles triangles include equilateral 
triangles, is much more convenient. So isosceles triangles should be 
defined as triangles that have *at least* two sides congruent.

Why is the inclusive definition more convenient? Well, consider a 
theorem like:

   If angle A = angle B in triangle ABC, then the triangle is 
   isosceles.

If you had the non-inclusive definition, you'd always have to be 
saying things like:

   If angle A = angle B but is not equal to angle C, then ...

or

   If ... then the triangle is isosceles or equilateral.

For the same reason, the definition of rectangle should include 
squares, the definition of parallelogram should include rectangles, 
and so on. The inclusive definition is almost always better.

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