1. Discrete Mathematics, Part IIIb CSE 2353 Fall 2007 Margaret H. Dunham Department of Computer Science and Engineering Southern Methodist University Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, MinnesotaSome slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Minnesota Some slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D.S. Malik and M.K. SenSome slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D.S. Malik and M.K. Sen

2. 2 Outline  Introduction  Sets  Logic & Boolean Algebra  Proof Techniques  Counting Principles  Combinatorics  Relations,Functions  Graphs/Trees  Boolean Functions, Circuits

3. 3 Functions

4. 4  Let A = {1,2,3,4} and B = {a, b, c, d} be sets  The arrow diagram in Figure 5.6 represents the relation f from A into B  Every element of A has some image in B  An element of A is related to only one element of B; i.e., for each a ∈ A there exists a unique element b ∈ B such that f (a) = b

5. 5 Functions  Therefore, f is a function from A into B  The image of f is the set Im(f) = {a, b, d}  There is an arrow originating from each element of A to an element of B  D(f) = A  There is only one arrow from each element of A to an element of B  f is well defined

6. 6 Functions

7. 7  Let A = {1,2,3,4} and B = {a, b, c, d}. Let f : A → B be a function such that the arrow diagram of f is as shown in Figure 5.10  The arrows from a distinct element of A go to a distinct element of B. That is, every element of B has at most one arrow coming to it.  If a 1, a 2 ∈ A and a 1 = a 2, then f(a 1 ) = f(a 2 ). Hence, f is one-one.  Each element of B has an arrow coming to it. That is, each element of B has a preimage.  Im(f) = B. Hence, f is onto B. It also follows that f is a one-to-one correspondence.

8. 8 Functions  Let A = {1,2,3,4} and B = {a, b, c, d, e}  f : 1 → a, 2 → a, 3 → a, 4 → a  For this function the images of distinct elements of the domain are not distinct. For example 1  2, but f(1) = a = f(2).  Im(f) = {a}  B. Hence, f is neither one-one nor onto B.

9. 9 Functions

10. 10 Functions  Let A = {1,2,3,4}, B = {a, b, c, d, e},and C = {7,8,9}. Consider the functions f : A → B, g : B → C as defined by the arrow diagrams in Figure 5.14.  The arrow diagram in Figure 5.15 describes the function h = g ◦ f : A → C.

11. 11 Functions

12. 12 Outline  Introduction  Sets  Logic & Boolean Algebra  Proof Techniques  Counting Principles  Combinatorics  Relations,Functions  Graphs/Trees  Boolean Functions, Circuits

13. 13 Outline  Introduction  Sets  Logic & Boolean Algebra  Proof Techniques  Counting Principles  Combinatorics  Relations,Functions  Graphs/Trees  Boolean Functions, Circuits

14. 14 Mathematical System

15. 15 Two-Element Boolean Algebra Let B = {0, 1}.

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17. 17 Boolean Expressions

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19. 19 Two-Element Boolean Algebra

20. 20 Minterm

21. 21 Disjunctive Normal Form

22. 22 Maxterm

23. 23 Conjunctive Normal Form

24. 24 Logical Gates and Combinatorial Circuits

25. 25 Logical Gates and Combinatorial Circuits

26. 26 Logical Gates and Combinatorial Circuits

27. 27 Logical Gates and Combinatorial Circuits

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