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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
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Discrete Mathematics, Part IIIb CSE 2353Fall 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
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
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
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 a1, a2 ∈ A and a1 = a2, then f(a1) = f(a2). 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.
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.
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.
Outline • Introduction • Sets • Logic & Boolean Algebra • Proof Techniques • Counting Principles • Combinatorics • Relations,Functions • Graphs/Trees • Boolean Functions, Circuits
Outline • Introduction • Sets • Logic & Boolean Algebra • Proof Techniques • Counting Principles • Combinatorics • Relations,Functions • Graphs/Trees • Boolean Functions, Circuits
Two-Element Boolean Algebra Let B = {0, 1}.
