Math 3325 Review for Exam 2 1 Definitions and Axioms

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 {x/R|x ∈ A} 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 {1, 2, 3}: (a) = (b) ≤ (c) 6= (d) S = {(1, 3), (2, 1)} (e) S−1, where S = {(1, 3), (2, 1)} (f) S ◦ S, where S = {(1, 3), (2, 1)}

2. Consider a relation R on R where for x, y ∈ R, xRy iﬀ x2 = y2. (a) Prove R is an equivalence relation (ER) on R. (b) For any x ∈ R, what is x/R?

3. Consider a relation R on R2 where for (a, b), (c, d) ∈ 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 = {a, b, c, d, e} and let S = {{a, b}, {b, c}, {a, b, e}}. 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, ∞) 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, ∞)

10. Let A = {1, 2, 3, 4} and let t = [2431 and s = [2314] be permutations of A. Find: (a) t ◦ s (b) s ◦ t (c) s−1