Parts of a Biconditional Statement

Date: 06/03/99 at 19:18:33
From: abdellah EL-IDRYSY
Subject: Propositional Calculus (the biconditional)

Dear all,

I came across the following arguments (in a book) involving the 
biconditional and the author's proof confused me. The author stated 
the following theorem:

  A quadrilateral is a square if and only if it is both a rhombus and 
  a rectangle 

and proved the theorem in two steps as follows:

Step 1 (the "if" part): Let Q be a quadrilateral which is both a 
rhombus and a rectangle. From the definition of a rhombus, all four 
sides are equal in length. From the definition of a rectangle, all 
four angles are right. A square is a four-sided figure in which all 
angles are right and all sides are equal. Therefore, Q is also a 
square.

Step 2 (the "only if" part): Let Q be a square. By definition, all 
four sides are equal, so Q is also a rhombus. Again by definition, all 
angles are right, so Q is a rectangle.

Until here I have followed the author's proof, but further he 
re-proves the same biconditional using "necessary" and "sufficient"; 
and there he wrote that:

step 1 (the "necessity" part): the same as "if" part in the last 
proof.

Step 2 (the "sufficiency" part): same as the "only if" part above.

I believe it's just the other way around: that is, "necessity" 
corresponds to "only if" and  "sufficient" corresponds to "if."

Could anybody be of some help? Am I right? More explanation to make 
the idea clearer is very welcome.

Many thanks in advance

Abdellah

Date: 06/08/99 at 15:01:49
From: Doctor Mike
Subject: Re: Propositional Calculus (the biconditional)

Maybe it would help to actually write out the sentences, but let's use 
"S" to mean Square and "R+R" to mean Rhombus and Rectangle. 

The first one is "S is a sufficient condition for R+R." This means 
that if you are given S, then you have "sufficient" (or "enough") 
information to prove R+R. That's exactly what was done for the "only 
if" part. 
  
Next, look at "S is a necessary condition for R+R". This should be 
the other direction, so let's see why. If you focus on the first part, 
"S is a necessary condition," then you see we really are talking about 
S being a necessary and logical consequence of something, namely R+R. 
Assuming R+R and proving S is what was done above in the "if" step.
    
Here is a summary. The following 4 sentences mean the same.
   
   S  ---->  R+R
   S  implies  R+R
   S  only if  R+R
   S  is sufficient for  R+R
   
The other direction also has the 4 similar variants.
   
   S  <----  R+R
   S  is implied by  R+R
   S  if  R+R 
   S  is necessary for  R+R

I hope this helps. 

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