MATH 2600H 2017 Winter Assignment 3 Due March 21, 2017 1. Section 7.2 #25. Let S be the set of all strings in a's and b's, and de...ne C : S ! S by C (s) = as; for all s 2 S: (a) Is C one-to-one? Prove or give a counterexample. (b) Is C onto? Prove or give a counterexample. 2. Section 7.3 #22. True of False? Given any set X and given any functions f : X ! X; g : X ! X; and h : X ! X; if h is one-to-one and f h = g h; then f = g. Justify your answer. 3. Section 7.4 #21. Give two examples of functions from Z to Z that are onto but not one-to-one. 4. Section 8.2 #2. R2 = f(0; 0) ; (0; 1) ; (1; 1) ; (1; 2) ; (2; 2) ; (2; 3)g is de...ned on the set A = f0; 1; 2; 3g : (a) Draw the directed graph. (b) Determine whether the relation is reexive. (c) Determine whether the relation is symmetric. (d) Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties. 5. Section 8.3 #12. A = f 4; 3; 2; 1; 0; 1; 2; 3; 4g : R is de...ned on A as follows: For all (m; n) 2 A; m R n , 5j m2 n2 : The relation R is an equivalence relation on the set A: Find the distinct equivalence classes of R. 6. Section 8.5 #3. Let S be the set of all strings of a's and b's. De...ne a relation R on S as follows: For all t 2 S; s R t , l (s) l (t) ; where l (x) denotes the length of a string x. Is R antisymmetric? Prove or give a counterexample. 7. Section 8.5 #20. Let S = f0; 1g and consider the partial order relation R de...ned on S S S; (a; b; c) R (d; e; f ) , a d; b e; and c f where denotes the usual \"less than or equal to\"relation for real numbers. Draw the Hasse diagram for R. 1 \f\f\f\f