How Do You Start The Proof?

The letters HDYSTP, used on many of the papers from February 25, returned on February 27, is an abbreviation for
       How Do You Start The Proof
(so it is often followed by a question mark:  HDYSTP?)

This is closely related to the subject of the quizzes given at the beginning and end of class on February 27.

The issue here is starting the proof of a universally quantified statement.

As we have discussed since we began talking about universally quantified if-then statementsin about the second week of class, there is a standard way to start the proof of a statement of the form
       “ For all  x in  S,  if  p(x)  then  q(x) ”
and its variations such as
       “ For all  x in  S,   q(x) ”
and
       “ For all  x,  if  p(x)  then  q(x) ”
and
       “ If  p(x)  then  q(x) ”  (implicit quantifier).

There are two reasons why I encourage this standard approach (I will not repeat here the approach here, because it has been stated in class approximately 12,346 times):

1. It gives you a place to start;
2. It can help you avoid errors which would result from starting in an INCORRECT way.

This standard approach was discussed generally in connection with Section 2, but also in several specific cases:

• It was discussed in connection with Sect. 5, when proving that a set  A is a subset of a set  B
  (the definition of  “A  is a subset of  B”  is the universally quantified statement  “For all  x in  A,  x is in  B”).
  
  
• It was discussed in connection with Sect. 6, when proving that a relation  R  is reflexive on a set  A
  (the definition of  “R  is reflexive on  A”  is the universally quantified statement  “For all  x in  A,  (x, x)  is in  R”).
  
  
• It was discussed in connection with Sect. 6, when proving that a relation  R  is symmetric (the definition of  “R  is symmetric”  is the universally quantified statement  “For all  (x, y)  in  R,  (y, x)  is in  R”).
  
  
• Similarly for transitive.

See also http://math.arizona.edu/~laetsch/323081/Lessons/PropertiesRelationsProofs.html, which is part of Lesson 20a, due February 29, and which you were encouraged to look at BEFORE doing the homework due February 27.

Last modified Feb 27, 2008 7:07 PM

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