Math 3325 Review for Exam 2 Exam 2 will cover the material in Chapter 3 and Sections 4.1-4.4 of Chapter 4 of A Transition to Advanced Mathematics (Seventh Edition) by Smith, Eggen, and St. Andre. Possible exercise types include true/false questions; statements of axioms, definitions, and major results; computational exercises; and exercises requiring theoretical arguments (proofs). Note that this serves as a review sheet but is by NO MEANS comprehensive regarding what will be on the exam. The best preparation will be to look through your notes, the book, and previous homework. 1 Definitions and Axioms You should be able to define and use the following 1. Relation 2. Digraph representation of a relation 3. Composition of Relations 4. Equivalence Relations 5. Equivalence Classes 6. Modular Congruence 7. Partition 8. Relationship between ECs and Partitions 9. Ordering Relations 10. Antisymmetric Relations 11. Partial Orderings and Posets 12. Upper Bounds and Suprema 13. Lower Bounds and infima 14. Linear Ordering 15. Well Ordering 16. Graphs 17. Walks, Paths, Reachability 18. Equivalence Classes in Graphs 19. Connected and Disconnected Graphs 20. Components of Graphs 21. Maximally connected subgraphs 22. Function; domain, codomain, range (image) 23. Equality of functions 24. Identity function IA : A ! A 25. Restriction of a function 26. Composition of functions 27. Increasing and decreasing functions on an interval I 28. Inverse function 29. Surjection, Injection 30. Bijection 2 Computational Techniques 1. Combinations, permutations 2. Generating partitions given an ER 3. Generating an ER given a partition 4. Determining domain, range of a function 5. Computing inverse of a function 6. Proving functions are injective or surjective 3 Know These Proofs 1. For every fixed positive integer m, ≡m is an equivalence relation on Z. 2. If R is an ER on a nonempty set A, then fx=Rjx 2 Ag is a partition of A. 3. Infimum and supremum for a set, if they exist, are unique 4. Let g; f be functions and assume g ◦ f is surjective. Prove that g is surjective. 5. Let g; f be functions and assume g ◦ f is injective. Prove that f is injective. 6. Set F be an injection. Show that F −1 is a function. 4 Sample Problems These are for your practice; solving these will not guarantee a strong performance on the exam. Studying your notes, the book, and old homework should will help. 1. Give the digraphs for these relations on the set f1; 2; 3g: (a) = (b) ≤ (c) 6= (d) S = f(1; 3); (2; 1)g (e) S−1, where S = f(1; 3); (2; 1)g (f) S ◦ S, where S = f(1; 3); (2; 1)g 2. Consider a relation R on R where for x; y 2 R, xRy iff x2 = y2. (a) Prove R is an equivalence relation (ER) on R. (b) For any x 2 R, what is x=R? 3. Consider a relation R on R2 where for (a; b); (c; d) 2 R2,(a; b)R(c; d) iff a2 − b = c2 − d. (a) Prove R is an ER on R2. (b) Describe in words what the elements in R2=R are. 4. Show that the relation R on N given by aRb iff a = 3kb for some integer k ≥ 0 is a partial ordering. 5. Let A = fa; b; c; d; eg and let S = ffa; bg; fb; cg; fa; b; egg. Find inf (S) and sup (S) with relation ⊆. 6. Verify these properties for the distance between vertices in a connected graph: (a) d(u; v) ≥ 0 (b) d(u; v) = 0 iff u = v (c) d(u; w) ≤ d(u; v) + d(v; w) 7. Prove that f(x) = x2 is increasing on [0; 1) and decreasing on (−∞; 0]. 8. Give an example of: (a) A surjection that is not injective (b) An injection that is not surjective 9. Prove that f(x) = x2 is a bijection for f :(−∞; 0] ! [0; 1) 10. Let A = f1; 2; 3; 4g and let t = [2431 and s = [2314] be permutations of A. Find: (a) t ◦ s (b) s ◦ t (c) s−1.