1. Relations Between Sets 2/13/121

2. Relations 2/13/122 Sam MaryCS20 EC 10 Students Courses The “is-taking” relation A relation is a set of ordered pairs: {(Sam,Ec10), (Sam, CS20), (Mary, CS20)}

3. Function: A → B 2/13/123 A B Each element of A is associated with at most one element of B. a b f(a) = b f AT MOST ONE ARROW OUT OF EACH ELEMENT OF A domain codomain

4. Total Function: A → B 2/13/124 A B Each element of A is associated with ONE AND ONLY one element of B. a b f(a) = b f EXACTLY ONE ARROW OUT OF EACH ELEMENT OF A domain codomain

5. A Function that is “Partial,” Not Total 2/13/125 R×R R f: R ×R → R f(x,y) = x/y Defined for all pairs (x,y) except when y=0! f domain codomain

6. A Function that is “Partial,” Not Total 2/13/126 R×R 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 f domain codomain

7. Injective Function 2/13/127 A B f domain codomain “at most one arrow in” ( ∀ b ∈ B)( ∃ ≤1 a ∈ A) f(a)=b

8. Surjective Function 2/13/128 A B f domain codomain “at least one arrow in” ( ∀ b ∈ B)( ∃ ≥1 a ∈ A) f(a)=b

9. Bijection = Total + Injective + Surjective 2/13/129 A B f domaincodomain “exactly one arrow out of each element of A and exactly one arrow in to each element of B” ( ∀ a ∈ A) f(a) is defined and ( ∀ b ∈ B)( ∃ =1 a ∈ A) f(a)=b

10. Cardinality or “Size” 2/13/1210 A B f domain codomain For finite sets, a bijection exists iff A and B have the same number of elements

11. Cardinality or “Size” 2/13/1211 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, 4 … …, -8, -6, -4, -2, 0, 2, 4, 6, 8 …

12. Cardinality or “Size” 2/13/1212 There are as many natural numbers as integers 0 1 2 3 4 5 6 7 8 … 0, -1, 1, -2, 2, -3, 3, -4, 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

13. An Infinite Set May Have the Same Size as a Proper Subset! 2/13/1213 0 1 2 3 4 5 0 1 2 3 4 5 HiltonSheraton Every room of both hotels is full! Suppose the Sheraton has to be evacuated ⋮ ⋮

14. An Infinite Set May Have the Same Size as a Proper Subset! 2/13/1214 0 1 2 3 4 5 0 1 2 3 4 5 HiltonSheraton Step 1: Tell the resident of room n in the Hilton to go to room 2n This leaves all the odd- numbered rooms of the Hilton unoccupied ⋮ ⋮

15. An Infinite Set May Have the Same Size as a Proper Subset! 2/13/1215 0 1 2 3 4 5 0 1 2 3 4 5 HiltonSheraton Step 2: Tell the resident of room n in the Sheraton to go to room 2n+1 of the Hilton. Everyone gets a room! ⋮ ⋮