et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈ S such that a + c = b. Show that if R is a partial order, then (i) 0 ∈ S (ii) for any a, b ∈ S, a + b ∈ S. You do not need to write a formal proof – a clear, concise and correct justification is sufficient.

et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈ S such that a + c = b. Show that if R is a partial order, then (i) 0 ∈ S (ii) for any a, b ∈ S, a + b ∈ S. You do not need to write a formal proof – a clear, concise and correct justification is sufficient.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

2. Define a relation R on Z as follows: a Rb if and only if 5 | (a² - b²). b) Find four integers which are equivalent to 3 (excluding 3 and -3).
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}.
Let L be a relation on R such that for all x and y in R, x L y if and only if x < y. Give a counter exampleto the statement ”L is an equivalence relation on R .
State whether each of the following statements is true or false. Then give an explanation, proof or counter example to justify your answer, The distinct equivalence classes of S form a partition of B. S is a partial order relation. If r and y are elements of B such that r y and rSy, then (9.2) S.
Let R be the relation on the set N0 of natural numbers given by the following rule: (n,m)∈R(n,m)∈R if and only if n−mn−m is a natural number. Is R reflexive?, symmetric?, antisymmetric?, transitive?, equivalence relation?, partial order? Prove your answer.
Theorem: Let R ⊆ A × A be a relation. Then R is transitive if and only if R ◦ R ⊆ R. Prove Theorem: show that R is transitive if and only if R ◦ R ⊆ R. No handwritten please
Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?
Let S be a nonempty subset of Z and let R be a relation defined on S by xRy if 3 | (x + 2y). If S = {−7, −6, −2, 0, 1, 4, 5, 7}, then what are the distinct equivalence classes in this case? Please show how you find them