Let A be a set and R and S be relations on A. Define the relation R+S on A by R+S = (R – S)U(S – R). Show that: (a) If A is countable, then so does R+ S. (b) but R+S is countable. Show that the converse does not hold; that is, give an example of a set A that is uncountable

Let A be a set and R and S be relations on A. Define the relation R+S on A by R+S = (R – S)U(S – R). Show that: (a) If A is countable, then so does R+ S. (b) but R+S is countable. Show that the converse does not hold; that is, give an example of a set A that is uncountable

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

Related questions

Q: Let R be the relation on the set of integers, s.t., aRb if and only if a=b or a= -b. Is R an…

Q: (a) Let R be a relation on the set Z × Z defined by the following rule: For all (a, b),(c, d) ∈ Z ×…

Q: Let W be the set of words in the English language. Define a relation R on W by W₁ R W₂W₁ and we have…

A: We are given that, w1 R w2 <=> w1 and w2 have a letter in common. Now, it is obvious…

Q: Is the following statement true or false? If R is any symmetric relation on a set A, then R is…

Q: Give an example of a relation on the set {a, b, c, d} which is symmetric and transitive, but not…

A: Any relation R from a set Ato A is reflexive if and only if for every x∈A , the pair (x,x)∈R Any…

Q: prove why each relation has or does not have the properties: reflexive, symmetric, anti-symmetric,…

A: The given relation R is as follows. R=x,y∈Z×Z:x is a multiple of y If x,y∈R, we denote x related to…

Q: Find the gradient field F of the function f. f(x, y, z) = x4e9x +y6₂⁹ OF = (4 + 9x)x3e9xi + 6y5z9;…

A: Given function f f(x,y,z) = x4e9x + y6z9 We need to find the gradient field F for the given…

Q: Determine whether or not the relation R on the set of all people is reflexive, netric,…

Q: Determine whether the relation R is a partial order relation on the set of positive…

Q: for the following relations on 2 the set A = {1, 2, 3}: R = {(1, 1), (1, 2), (1, 3), (3, 3)},…

A: A relation R is an equivalence relation if it is: i) Reflexive: if aRa for all a in A ii) Symmetric:…

Q: Let R = { ( a, b ), ( a, d ), ( b, d ), ( c, a ), ( c, c ), ( d, b ), ( d, c ) } be a relation on…

A: Given: R=a,b,a,d,b,d,c,a,c,c,d,b,d,c The relation R2 contains all ordered pairs (x, z) if (x, y)…

Q: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) E…

Q: set {(a,b),(a, a), (c, a), (b, b)} (a). Let X = {a,b,c} be sets. is a relation on X and find its…

A: Answer and explanation is given below..

Q: Decide whether each of the following relations is an equivalence relation or not. {(a, b) | gpa of…

A: To prove equivalence relation, we have to prove that (i) the relation is reflexive, (ii) the…

Q: b) E Rif and only if a has the same first name as b. (Check all that apply.) Check All That Apply…

A: We need to check whether the given relation is reflexive, symmetric, transitive, antisymmetric. here…

Q: Let A = {1, 2, 3, 4, 5} and R be the relation defined by a R b if and only if a < b. (a) Compute R²…

A: use the concept of composition R2=RoR R3=R2oR

Q: Let R be a relation defined on the set of integers as follows: a R b iff 6 divides (a - b). R is…

Q: A relation R is called atransitive if aRb and bRc implies cRa. Show that R is reflexive and…

A: Given a relation R is called atransitive if aRb and bRc⇒cRa Now shows in the following that R is…

Q: What is the reflexive closure of the relation R = {(a , b) I a < b} on the set of integers

Q: Let A be a set and R and S be relations on A. Dene the relation R + S on A by R + S = (R-S) […

A: Let A be a set and R and S be relations on A. Define the relation R + S on A by R + S = (R-S) ∪…

Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by R=1,1,2,2,3,31,2,2,1 And…

Q: 2. Let R be a relation defined on the set of integers Z where a R b if a +b is even. Determine…

Q: Let R be a relation defined on Z as follows: (a,b) is an element of R if and only if the square of a…

Q: Let R be the relation on a set A = {a, b, c, d} defined by R= {(a, a), (a, b), (b, b), (b, c), (c,…

A: If R is a relation on a set A, then R^2 is the set of ordered pairs (a, c) such that there exists an…

Q: For each relation R below, determine whether it is an equivalence relation on the given set A. Also,…

Q: Define the relation R on 72 = Z x Z by (a, b) R (c, d) if a + d = b+ c. Prove that R is ar…

Q: Determine if the relation described is an equivalence relation on the given set. Let S = Z+, a ~ b…

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

Q: 1. Let A = {2, 3, 4, 5, 6} and B = {10, 11, 12, 13). Define a relation R on (A x B) as follows: a Rb…

Q: For the set A = {are, bot, cab, era)}, define the relation R on A to be: (x,y) eR if and only if…

A: As per the condition, in every order pair there should be only 1 letter in common.

Q: Let R be the relation on a set A = {a, b, c, d} defined by R = {(a, a), (a, b), (b, b), (b, c), (c,…

A: The given relation on a set is defined as follows..We need to find .

Question

Let A be a set and R and S be relations on A. Define the relation R+ S on A by
R+S = (R – S)U (S – R).
-
Show that:
(a)
If A is countable, then so does R+ S.
Show that the converse does not hold; that is, give an example of a set A that is uncountable
(b)
but R+ S is countable.
(c)
Prove or disprove: If R and S are partial orders on A, then so does R+ S.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

part a

VIEW

part b

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Relations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Is the relation aRb → ab < 0 an equivalence relation on the set Z? Justify your answer.
Let R be the relation on the set of ordered pairs of positive integers such that ((a,b), (c,d)) e R if and only if a + d = b + c. Determine whether R is reflexive, symmetric and transitive.
Let the relation R on the set of ordered pairs of positive integers be defined as (a, b) R (c, d) a, b, c, d are positive numbers. Show that R is an equivalence relation. a + d = b + c, where
Given a set A = {1, 2, 3, 4}. Define a relation R on A as: R = {(x, y) | x - y is even} List the ordered pairs of the relation R. Is the relation R reflexive, symmetric, and transitive? Justify your answers.
Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R = IdA, then S is surjective. If the assertion is not true, find additional assertions that will make it true.
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
part b and c
please answer letter g
Prove or disprove.
Give examples of relations R and S on Z such that (a) Both R and S are not equivalence relations but R+ S is. (b) Both R and S are not partial order relations but R+ S is. Show that if both R and S are equivalences relations, then R+S is not an equivalence relation.
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c,d)) = R if and only if ad = bc. Show R is an equivalence relation.