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 Outline Introduction Sets Logic & Boolean Algebra Proof Techniques Counting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits 3 Functions 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 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 Functions 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 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 Functions 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 The arrow diagram in Figure 5.15 describes the function h = g f : A C. 11 Functions 12 Outline Introduction Sets Logic & Boolean Algebra Proof Techniques Counting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits 13 Outline Introduction Sets Logic & Boolean Algebra Proof Techniques Counting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits 14 Mathematical System 15 Two-Element Boolean Algebra Let B = {0, 1}. 16 17 Boolean Expressions 18 19 Two-Element Boolean Algebra 20 Minterm 21 Disjunctive Normal Form 22 Maxterm 23 Conjunctive Normal Form 24 Logical Gates and Combinatorial Circuits 25 Logical Gates and Combinatorial Circuits 26 Logical Gates and Combinatorial Circuits 27 Logical Gates and Combinatorial Circuits 28 29 30 31 32 33