WEEK 4  
M: Indexed systems. Basic formula with indexed systems. Section 3: Induction and recursion.
      The induction principle. Variation starting at a. Examples.  
       2.9. Proposition. (a) A_i is a subset of the union. The intersection is a subset of A_i. 
                                   (b) Distributivity. 
                                   (c) De Morgan's laws.
       3.1. Example. The sum of the first n squares. 
       3.2. Example. 16ⁿ - 11ⁿ is divisible by 5. (Complete next time.)  
      PROBLEMS FOR DISCUSSIONS:  Book, p. 59 Exercise 60, 61, 63, 64, 67, 72
W: Completion of Example 3.2. More examples of induction. Factorial. Combination number.
      3.3. Example. 1 + 1/(\sqrt{2} + ······· + 1/(\sqrt{n} >  \sqrt{n}  
      3.4. Example.  There are n² unit triangles inside a regular triangle with side of length n units.
      3.5. Definition.  n!  
      3.6. Definition. Combination number "n over k".
      3.7. Proposition.  "n over k" + "n over (k-1)" =  "n+1 over k". (Proof next time.)
     PROBLEMS FOR DISCUSSIONS: No new problems. Complete the problems on unions and intersections
                                                                     of indexed systems assigned previously.  
F: Proof of Prop. 3.7. Binomial formula. Strong induction.
     Example 3.8. Binomial formula -- proof by induction.
     Example 3.9. Every natural number larger than 1 is divisible by a prime. Proof by strong induction.
     PROBLEMS FOR DISCUSSIONS: Book, p.73 Exercise 1egjknopq, 2