Discrrete mathematics for computer science 10relations

Relations Between Sets 2/13/12 Relations Students Courses Sam EC 10 Mary CS20 The “is-taking” relation A relation is a set of ordered pairs: {(Sam,Ec10), (Sam, CS20), (Mary, CS20)} 2/13/12 Function: A → B AT MOST ONE ARROW OUT OF EACH ELEMENT OF A domain f codomain A B Each element of A is associated with at most one element of B 2/13/12 a⟼b f(a) = b Total Function: A → B EXACTLY ONE ARROW OUT OF EACH ELEMENT OF A domain f codomain A B Each element of A is associated with ONE AND ONLY one element of B 2/13/12 a⟼b f(a) = b A Function that is “Partial,” Not Total domain f R×R codomain R f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! 2/13/12 A Function that is “Partial,” Not Total domain f R×R codomain R f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! Or: f is a total function: R×(R-{0})→R 2/13/12 Injective Function “at most one arrow in” domain f A codomain B (∀b∈B)(∃≤1a∈A) f(a)=b 2/13/12 Surjective Function “at least one arrow in” domain f A codomain B (∀b∈B)(∃≥1a∈A) f(a)=b 2/13/12 Bijection = Total + Injective + Surjective “exactly one arrow out of each element of A and exactly one arrow in to each element of B” domain f A codomain B (∀a∈A) f(a) is defined and 2/13/12 (∀b∈B)(∃=1a∈A) f(a)=b Cardinality or “Size” For finite sets, a bijection exists iff A and B have the same number of elements domain A 2/13/12 f codomain B 10 Cardinality or “Size” Use the same as a definition of “same size” for infinite sets: Sets A and B have the same size iff there is a bijection between A and B Theorem: The set of even integers has the same size as the set of all integers [f(2n) = n] …, -4, -3, -2, -1, 0, 1, 2, 3, … …, -8, -6, -4, -2, 0, 2, 4, 6, … 2/13/12 11 Cardinality or “Size” There are as many natural numbers as integers 8… 0, -1, 1, -2, 2, -3, 3, -4, … f(n) = n/2 if n is even, -(n+1)/2 if n is odd Defn: A set is countably infinite if it has the same size as the set of natural numbers 2/13/12 12 An Infinite Set May Have the Same Size as a Proper Subset! ⋮ 5 4 3 2 1 0 Hilton 2/13/12 ⋮ Every room of both hotels is full! Suppose the Sheraton has to be evacuated Sheraton 13 An Infinite Set May Have the Same Size as a Proper Subset! ⋮ ⋮ 5 4 Step 1: Tell the resident of room n in the Hilton to go to room 2n 3 2 This leaves all the odd-numbered rooms of the Hilton unoccupied 1 0 Hilton 2/13/12 Sheraton 14 An Infinite Set May Have the Same Size as a Proper Subset! ⋮ ⋮ 5 4 Step 2: Tell the resident of room n in the Sheraton to go to room 2n+1 of the Hilton 3 2 1 0 Hilton 2/13/12 Everyone gets a room! Sheraton 15 ... → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! 2/13/12 A Function that is “Partial,” Not Total domain f R×R codomain R f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except... “Size” For finite sets, a bijection exists iff A and B have the same number of elements domain A 2/13/12 f codomain B 10 Cardinality or “Size” Use the same as a definition of “same size” for infinite