Parts of a Biconditional StatementDate: 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/ |
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