Discrete Mathematics Part IIIb CSE 2353 Fall 2007

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, Minnesota • Some slides are companion slides for Discrete Mathematical Structures: Theory and Applications by D. S. Malik and M. K. Sen

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

Functions 3

Functions ¡ ¡ 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 4

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 5

Functions 6

Functions ¡ ¡ ¡ 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. 7

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. 8

Functions 9

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. 10

Functions 11

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

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