First posted Feb 5

Last modified Feb 9, 2008 8:46 AM

• In Lesson 1, you were asked to explain why the sentence “All integers are not even“ is ambiguous. Most or all students were able to recognize the ambiguity.
  
  The point was reiterated in Lesson 2.
  
  Therefore, it should be clear that, in Exercise 2.4(a), the correct negation of
  “Some basketball players ... are short”
  will NOT be
  “All basketball players ... are not short”.
  
  The latter sentence suffers from the same ambiguity as the “All integers are not even” sentence in Lesson 1.
  
  See the Discussion of Lesson 1, posted on or before Jan 27!
  
  
• A very common error on recent homework (and on the in-class quiz Monday) was to do a proof backwards, an issue we have discussed since the first day of class. This usually consisted of starting an algebraic argument in your proof with the statement you are trying to prove.
  
  E.g., if  z = y-x  and you are trying to prove that  x+z = y,  you don't START the verification by assuming that  x+z = y.
  
  Similarly, if you are trying to prove that THERE EXISTS  z  such that  x+z = y,  you don't START the proof by assuming that there exists  z  such that  x+z = y  and then finding that  z  must be  y-x.
  
  In a couple of cases, there were “proofs” which ended with  “y = y”,  a telltale sign that something is probably being done backwards.
  
  
• In Exercise 2.2(b) and (c), from a previous homework assignment, two points were emphasized:
  
  (b) “If a variable is used in the antecedent of an implication without being quantified, ... ”  [you can read the rest for yourself if you don't remember it].
  As emphasized in class, this point is particularly important when negating a statement with an implicit universal quantifier.
  
  (c) [see next bullet]
  
  Exercise 2.18 is an example of (b), a statement with a (more or less) implicit universal quantifier -- the x and y in this statement are universally quantified.  This means that, when negating the statement, you need an existential quantifier on the x and  y.  This is very important, as pointed out in class.  The negation without the existential quantifier on x and  y is either incorrect or, more likely, meaningless.
  
  Simple example:

  “If  x is a real number, then  x² + 1 > 0.”

  Negation?

  “x is a real number and  x² + 1  is not greater than 0.”

  This is not a correct negation.  [THINK ABOUT IT. What does this “statement” mean? Who is this  x  we are talking about?] The original statement is a universally quantified statement (for all x, ... ). To be both meaningful and correct, the negation should be

  “THERE EXISTS  x  such that  x  is a real number and ... .”

  (or some equivalent form).
  
  
• Continuing with Exercise 2.2 ...
  
  (c) “The order in which quantifiers are used affects the truth value.”
  [This is approximately true; it would be better stated as, “The order in which existential and universal quantifiers are used may affect the truth value of a statement containing such quantifiers.” ]
  
  The book gives at least one example of this, which many of you noted in your solution to Ex. 2.2(c).
  
  Yet, when doing the homework on problems like Exercise 2.5, this point was usually ignored.
  
  Example:
         “There exists x such that for all  y,  y > x ”
  is false.
         “For all  y, there exists x such that  y > x ”
  is true.
  
  Similarly (cf. Exercise 2.5(e)),
         “There exists x such that for all  y and for all z ... ”
  is different from
         “For all  y and for all z, there exists x such that ... ”.
  These are different statements and probably require different proofs (if they are true).
  
  In particular, the proof of the first statement requires that you start with an  x before knowing what  y  and  z  are.
  
  

MORAL

When doing Lesson n,
     don't forget what you have done in Lesson k (for k < n)!

In fact, as in the examples cited above, very often the whole point of a particular Lesson k was to prepare you for some Lesson n (with n > k).

Back to Course Home Page