WEEK 9
M: Basic facts about composition and inverses of relations. Functions. 
      5.9. Proposition. (a)  (R^-1)^-1 = R 
                                       (b) R^-1 o R contains  id_dom(R)   and   R o R^-1  contains id_rng(R)  
                                       (c) (T o S) o R = T o (S o R) 
                                       (d) (S o R)^-1 = R^-1 o S^-1
     5.10. Definition. Function. Function from A to B, f : A --> B.
     PROBLEMS FOR DISCUSSIONS: No new problems. Complete the previously assigned problems.
W: Remarks concerning functions. Injectivity, surjectivity, bijectivity. Sequences.
      5.11. Definition. Injectivity, surjectivity, bijectivity.
      5.12. Definition. Sequences of length n, infinite sequences.
      PROBLEMS FOR DISCUSSIONS:  * Book, p.156 Exercise 4, 6 , 8, 9, 11
                                                                      * Prove: (a) composition of two injective functions is ijnective 
                                                                                      (b) composition of two surjective functions is surjective
                                                                                      (c) composition of two bijective functions is bijective
                                                                                      (d) if f : A --> B is bijective then f^-1: B --> A is bijective
                                                                      * Book, p. 161 Exercise 23, 24  
F: Examples on maps/injectivity/surjectivity. Motivation for the notion of equinumerosity.
     5.13. Proposition. f-1 is a function iff f is injective; dom(f-1)=rng(f).
     5.14. Examples. (a) f: Z ×  Z -->  Z defined by f(<a,b>) = g.c.d.(a,b)
                                  (b) f: P(A) --> P(A) defined by f(X) = A -X  .
                                      Proof that A has the same number of even and odd subsets.
    PROBLEMS FOR DISCUSSIONS: No new problems. Finish all old problems.