First, see if students have any questions and answer them, if there are 
any. 

 - Go over the materials on page 32. That is recall the definitions of:
     . graph, binary tree, full binary tree
     . root, internal node, parent, child, leaf,  
   They'll need these for assignment 1.

Examples of wrong inductive proofs:

 - Example 1.8 on page 31 of the notes.

 - Exercise 13 on page 58 of the notes. First state the problem clearly 
   and let them think about it for a few minutes. 
   
   Here is the solution:
   
   The error is in the inudction step. The proof contains "Since i - 4 < i
   by induction hypothesis we can make postage of exactly i - 4 cents...".
   This statement is true only if i - 4 >= 4, because the induction 
   hypothesis applies to values j such that 4 <= j < i.
   Therefore, to have i - 4 >= 4 we need to have i >= 8. But the only
   assumption on i that we have is i >= 4. Therefore, there is a gap 
   between the base case and the induction step and the proof is false.


In the unlikely case that you have extra time go over this example:

 - Exercise 7 on page 57 of the notes.

   Here is a solution:
   
   We define predicate S(n) to be: 
                                n
      "For any integer m >= 2, sum m^t = (m^{n+1} - 1)/(m - 1)"
                               t=0
   We prove by induction that S(n) is true for all integers n >= 1.
   Base: If n = 1 then,
      n         1
     sum m^t = sum m^t = m^0 + m^1 
     t=0       t=0
                       = 1 + m = (m + 1)(m - 1)/(m - 1)
                       = (m^2 - 1)/(m - 1)
                       = (m^{n+1} - 1)/(m - 1).

   Ind. Hyp: For an arbitrary integer k >= 1, assume that S(k) is true.
   Ind. Step: We want to prove S(k + 1) is true.
     k+1        k
     sum m^t = sum m^t + m^{k + 1} 
     t=0       t=0
             = (m^{k + 1} - 1)/(m - 1) + m^{k + 1}     By Ind. Hyp.
             = (m^{k + 1} - 1)/(m - 1) + m^{k + 1}(m - 1)/(m - 1)
             = (m^{k + 1} - 1 + m^{k + 2} - m^{k + 1})/(m - 1)
             = (m^{k + 2} - 1)/(m - 1),
     as wanted.