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Relations Between S ets

Relations Between S ets. 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)}. Function: A → B. AT MOST ONE ARROW OUT OF EACH ELEMENT OF A. f.

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Relations Between S ets

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  1. Relations Between Sets

  2. 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)}

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

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

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

  6. A Function that is “Partial,” Not Total f domain codomain 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

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

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

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

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

  11. 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, 4 … …, -8, -6, -4, -2, 0, 2, 4, 6, 8 …

  12. Cardinality or “Size” 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 infiniteif it has the same size as the set of natural numbers

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

  14. An Infinite Set May Have the Same Size as a Proper Subset! ⋮ ⋮ 5 5 4 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 4 3 3 2 2 1 1 0 0 Hilton Sheraton

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

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