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 34.3, Problem 2E
Program Plan Intro
To show that
If
and
Then
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Consider the relation R = {(a, d), (b, a), (b, c), (c, b), (c, c), (d, a)} on the set {a, b, c, d}.
Find the reflexive closure of R.
' Find the symmetric closure of R.
Find the transitive closure of R.
a. The relation x ≡ y if and only if x mod 4 == y mod 4 is an equivalence relation. Use this equivalence relation to partition the set {2, 4, 5, 6, 9, 23, 24, 31, 37} into equivalence classes.
b. How many substrings “aab” are there in wwRw, where w = aabbaab?
c. Prove that (LR)* = (L*)R for all languages L.
a) Define a relation R on R (the set of all real numbers) as follows: For all x, y E R, xRy →
x < y. Is R reflexive, symmetric and transitive?
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- The domain for the relation R is the set of all integers. For any two integers, x and y, xRy if x evenly divides y. An integer x evenly divides y if there is another integer n such that y = xn. (Note that the domain is the set of all integers, not just positive integers.) Symmetric Transitive Anti-symmetric Reflexivearrow_forwardLet S = {1,2,3}. Let be the relation on S x S defined by (a, b) ≤ (c,d) a ≤ c and b>d (a) Show that is a partial order on S x S. (b) Draw the Hasse diagram of the relation. (c) List all elements of S x S which are incomparable with (2,3).arrow_forwardLet A equal the set of all strings of 0's, 1's, and 2's that have length 4 and for which the sum of the characters in the string is less than or equal to 2. Define a relation R on A as follows: For every s, t E A, s Rt + the sum of the characters of s equals the sum of the characters of t. It is a fact that R is an equivalence relation on A. Use set-roster notation to list the distinct equivalence classes of R. (Enter your answer as a comma-separated list of sets.)arrow_forward
- Let A be a non-empty funute set and let S = P(A). Define relation R ⊆ S × S by: xRy ⇔ |x| = |y|, where | · | denotes the set cardinality.(i) Prove that R is an equivalence relation. (ii) Find all equivalence classes for the case A = {1, 2, 3}.arrow_forwardLet A={1,2,3} and B={4,5,6}. The set X={(1,4),(3,6),(2,4),(1,5)} is: O a relation between from A to B, but NOT a function from A to B O a set that has nothing to with A and B O equal to AxB O a function from A to Barrow_forwardFEE*, define o =T if for all w E E*, ow E L(M) if anc M). Show that = is an equivalence relation on D".arrow_forward
- Given the sets A = {a,b, c, d}, B = {0, 1, 2, 3} construct a relation R such that R is an injective function and R-1 is a surjective function. Represent R and R-1 with graphs.arrow_forwardGiven: 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_forwardConsider the relation R on Z? given by: R = {((a, b), (c, d)) : a +d = c+b} Prove that R is an equivalence relation, and determine 3 elements of [(1,3)], the equivalence class of (1, 3).arrow_forward
- 1. Let A = {3, 4, 5} and B = {4, 5, 6} and let S be the "divides" relation. That is, for all (x, y) ≤ A × B, xSy ↔ x|y. State explicitly whic ordered pairs are in S and S-1arrow_forward3. Given a relation R( A, B, C, D) where FDs ans CKs are such : FDs CKs AB -> C, BC -> D, CD -> A АB and BC А -> ВCD, C ->A A,C А->вс, С->BD, В->А No Clue Check the Highest Normal Form.arrow_forward-------If R is reflexive, symmetric and transitive then the relation is said to be Floor(2.4) + Ceil(2.9) is equal to Let f and g be the function from the set of integers to itself, defined by f(x) = 2x + ---- 1 and g(x) = x ^2. The composition of fog (-2) is The power set of (a} is -----arrow_forward
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