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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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!).
pAp
Consider the partal order on the set X(a. b.c.d.e.1)with the following relations
aくも
くe
もくe
もく。
esd
dse
PATE
pVT
PATE
PVTE
p VT
Which digraph represents this partialy ordered sel?
Given: 1. ∀s1,s2 Subset(s1,s2) ⇔ [∀x Member(x,s1) ⇒ Member(x,s2)].Prove: H. ∀s1,s2,s3 [Subset(s1,s2) ∧ Subset(s2,s3)] ⇒ Subset(s1,s3).
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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- 1. Consider the binary relation ⊆: ”is subset of” in the domain of sets. Prove that ⊆ is a partial order.2. Given the set V2 = {a, b, c}, prove that (P(V2), ⊆) is a poset.3. Create a Hasse Diagram to visualize (P(V2), ⊆), and identify maximal and minimal elements.arrow_forwardThe format P(X) represents as a power set of given "X", (of all X's subsets) Now, assume there are two sets A= {q, w, e, r} and B = {w, r, f}. (a) What are the elements of P(A) ∩P(B )?(b) What is the cardinality of P(A ∪B )?(c) What is |P(A) ∪P(B )|?arrow_forwardProve or disprove the following statement: If f is a bijection be- tween two finite sets, does that mean that the sets must have the same size?arrow_forward
- A= B if and only if : * A is subset from B and B is not subset from A A is not subset from B and B is subset from A A is subset from B and B ist subset from A if p q and -p V q is logically.. equivalent not equivalent tautologically Other:arrow_forwardProve that equivalences of sets are transitive, for example if X ~ Y and Y ~ Z then X ~ Z.arrow_forwardIs A ⊆ B if A = {8, 10, 12, 14} and B = {8, 10, 12, 16, 20}?arrow_forward
- 4. Problem Let T be a set and partially order P(T) by inclusion. Let C ≤ P(P(T)) be a chain and let F be a finite subset of Uuec U Prove that there is some U E C such that F C U. Hint: Clearly this is an induction problem.arrow_forwardIf P = { a,b,c,d,e } and Q = { a,q,r,s,e }, how many members has (P U Q) ?arrow_forwardProve: If a set A is a subset of the empty set, then A itself has to be an empty set.arrow_forward
- Briefly show if it is true or false and explain. A,B, and C are sets. If A ∈ B and B ∈ C, then is it impossible that A ∈ C?arrow_forwardDefine poset If a poset has a least element, then prove that it is unique. The following is the Hasse diagram of a partially ordered set. Verify whether it is a Lattice or not. e с b a darrow_forwardWrite Algorithm for Membership TestingA group G operating on the function f = 1, 2,..., n; a permutation of f = 1, 2,....., n; a base and strong energising set for G; and Schreier vectors v I 1 I k, for the stabilising chain; are the inputs.The function is in group(p: permutation; I 1..k+l): boolean; (* return true if the permutation p is in the group G I outputs a boolean value indicating if g G.arrow_forward
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