Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter B.2, Problem 1E
Program Plan Intro
To prove that the subset relation,
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Given set A = {a, b, c, d} show the equivalence relation, which contains eight ordered pairs, that induces this partition of A : {{a, c}, {b, d}}.
Prove that cardinality is reflexive, i.e., show that a set has the same cardinality as itself.
Prove that cardinality is transitive, i.e., show that if |A| “ |B| and |B| “ |C| then |A| “ |C|.
M is the matrix representation of a relation R on A. A has n elements. M is a n x n matrix. M contains how many 1s and 0s if R is a rooted (directed) tree?
Chapter B Solutions
Introduction to Algorithms
Ch. B.1 - Prob. 1ECh. B.1 - Prob. 2ECh. B.1 - Prob. 3ECh. B.1 - Prob. 4ECh. B.1 - Prob. 5ECh. B.1 - Prob. 6ECh. B.2 - Prob. 1ECh. B.2 - Prob. 2ECh. B.2 - Prob. 3ECh. B.2 - Prob. 4E
Ch. B.2 - Prob. 5ECh. B.3 - Prob. 1ECh. B.3 - Prob. 2ECh. B.3 - Prob. 3ECh. B.3 - Prob. 4ECh. B.4 - Prob. 1ECh. B.4 - Prob. 2ECh. B.4 - Prob. 3ECh. B.4 - Prob. 4ECh. B.4 - Prob. 5ECh. B.4 - Prob. 6ECh. B.5 - Prob. 1ECh. B.5 - Prob. 2ECh. B.5 - Prob. 3ECh. B.5 - Prob. 4ECh. B.5 - Prob. 5ECh. B.5 - Prob. 6ECh. B.5 - Prob. 7ECh. B - Prob. 1PCh. B - Prob. 2PCh. B - Prob. 3P
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- Implement the following Racket functions: Reflexive-Closure Input: a list of pairs, L and a list S. Interpreting L as a binary relation over the set S, Reflexive-Closure should return the reflexive closure of L. In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X that contains R. (https://en.wikipedia.org/wiki/Reflexive_closure) Examples: (Reflexive-Closure '((a a) (b b) (c c)) '(a b c)) ---> '((a a) (b b) (c c)) (Reflexive-Closure '((a a) (b b)) '(a b c)) ---> '((a a) (b b) (c c)) (Reflexive-Closure '((a a) (a b) (b b) (b c)) '(a b c)) ---> ((a a) (a b) (b b) (b c) (c c)) (Reflexive? '() '(a b c)) ---> '((a a) (b b) (c c)) You must use recursion, and not iteration. You may not use side-effects (e.g. set!).arrow_forwardProve that equivalences of sets are transitive, for example if X ~ Y and Y ~ Z then X ~ Z.arrow_forwardSuppose that a B+-tree index on building is available on relation departmentand that no other index is available. What would be the best way to handle thefollowing selections that involve negation?a. σ¬ (building < “Watson”)(department)Practice Exercises 791b. σ¬ (building = “Watson”)(department)c. σ¬ (building < “Watson” ∨ budget <50000)(department)arrow_forward
- Let S be a set and let C = (π1, π2,...,πn) be an increasing chain ofpartitions (PART(S), ≤) such that π1 = αS and πn = ωS. Then, the collection HC = ni=1 πi that consists of the blocks of all partitions in the chain is a hierarchy on S.arrow_forwardLet A be a non-empty set, ≃ ⊆ A × A an equivalence realtionship and ⪯ ⊆ A × A a partial order, both over A. Consider the quotient set: A / ≃, and define the following relation ≪ ⊆ (A/ ≃) × (A/ ≃) where (S1, S2) ∈ ≪ if, and only if, there is an a ∈ S1 such that for all b ∈ S2, a ⪯ b. 1.) How to show that ≪r is is a partial order over A/ ≃ where ≪r is the reflex cause of ≪. 2.) Is it true that A has a minimal element according to ⪯ if, and only if, A/ ≃ has a minimal element according to≪r ? How to show if it is true or false (in such case suggest a counterexample)arrow_forwardProve that the two properties of the hierarchy function allow only trees and single nodes as organizations of objectsarrow_forward
- For a given relation R on the set A, let M stand for the matrix representation of R. Let's pretend there's a set A with n unique items. Therefore, M would be an n-by-n matrix. If R is a rooted (directed) tree, how many 0s and 1s does M have?arrow_forwardSuppose that a B+-tree index on building is available on relation departmentand that no other index is available. What would be the best way to handle thefollowing selections that involve negation? σ¬ (building < “Watson” ∨ budget <50000)(department)arrow_forwardLet M, represent the matrix representation of a certain relation R on the set A. Consider a set A with n distinct elements. M, would therefore be a n x n matrix. How many 1s and 0s will M contain if R is a rooted (directed) tree?arrow_forward
- Given: Relation R = {(a,a), (a.b), (b,a). (b,b). (c,c)} and Set S = {a, b, c} Complete Relation R to be Transitive with Set Sarrow_forwardLet R ⊆ A × A be a a binary relationship defined in a non-empty set A. Let B be any set so that B⊆ A. Define the set Rb as RB := R ∩ (B × B). 1.) How to show that if R is reflexive, symmetric and transitive then Rb also is. 2.) Is the opposite true? Demonstrate truth or falseness. 3.) If R is antisimetric, is Rb antisimetric too?arrow_forwardLet A = {2,3,4,5} and let R be a relation on A such that xRy if and only if x+y>=4. (a) List the elements of R. (b) Draw a directed graph representation of R (c) is R reflexive? is R symmetric? Is R transistive? Justify your answer to each.arrow_forward
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