Math 314: Discrete Math Corrections and Comments

Corrections and Comments on the Book

• In Section 1.2 the authors write expressions like
  A --> B --> C
  which is a sequence of implications. This can be misleading! When we use the logical operation --> to combine propositions into a compound proposition, if we write "p --> q --> r" we have written something meaningless! We can only use --> to combine two propositions, not three. There is no definition (in the book or anywhere else) of what it would mean to combine three at once. If we want to combine three, we have to do them two at a time. What the authors wrote here is technically nonsense. You probably know what they meant, but it should be written as
  (A --> B) and (B --> C) .
  
  

  "Either p or q" could mean a lot of things in ordinary speech. In logic we mean: "p or q".
  
  

  The proper logical placement of quantifiers is: always in front of the statement that is quantified. For instance, ``x = x for every number x'' in logically correct form would be ``(For every real number x) x = x.'' The reason for this is to avoid ambiguous statements.
  
  

  The list of basic laws of the algebra of logic (pp. 22-23) should include the Absorption Laws:

  p and (p or q) <==> p

  p or (p and q) <==> p
  (which are proved in Exercise 1.4.3).
  
  

  P. 48, definition of equality of ordered pairs: there is a TERRIBLE ERROR. The correct definition is:
  
  

  (a₁, b₁) = (a₂, b₂) if and only if a₁ = a₂ and b₁ = b₂.
  
  

  In Definitions 3.1.3, you should add the name one-to-one correspondence as a commonly used name for ``bijective function''.

  P. 107: The method in the lower half of the page (through Problem 10), that uses just a table of numbers in three columns, is NOT ACCEPTABLE. Do not use it in homework. It will not be accepted on tests or quizzes.

  P. 121: The wiggly equals sign does not mean ``approximately''. It means ``approximately equals''.

  P. 128: We should have, as a companion to Corollary 4.4.4, the theorem I stated in class:

  Theorem. a is congruent to b modulo n <==> a and b have the same remainder upon division by n.
  You may use this theorem in addition to the ones in the book.