Let A = {1, 2, 3} and let ρ be a relation on P (A) × Z that maps a set to its cardinality, e.g. ({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A) What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written justification as well as the set B itself.

Let A = {1, 2, 3} and let ρ be a relation on P (A) × Z that maps a set to its cardinality, e.g. ({1, 2}, 2) ∈ ρ. (The notation P (A) denotes the powerset of A) What is the set B ⊂ Z with the fewest elements such that ρ ⊂ P (A) × B? Provide a written justification as well as the set B itself.

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

Let U = {1, 2, 3, 4, 5} be the universal set. Let A = {1, 2, 3}, B = {1, 3, 5}, and C = {2, 4, 5}.Determine: |A × (B × C)|
Define a relation R on the set {1, 2, 3, 4} as follows: T = {(1, 4),(2, 3),(2, 4),(4, 1),(2, 1),(1, 2),(3, 2)} (a) Is R symmetric? Justify your answer. (b) Is R transitive?
For the set A = {are, bot, cab, era}, define the relation R on A to be: (x,y) e R if and only if word x and word y share one letter only. For example (are, cab) is in R because those words share one letter only ("a"). But (are, era) is NOT in R because those words do NOT share one letter only, they share 3 letters. Also (bot, era) is NOT in R since they share no letters at all. In cach part, answer yes or no. If yes, then explain why in terms of this specific relation about sharing letters (not just by stating the generic definition of the property). If you say a) Is R reflexive? b) Is R symmetric? c) Is R antisymmetric? d) Is R transitive?
Consider the set S = {A, B, C, D}: For each of the following relations on S, determine whether it is a partial order, a total order, an equivalence relation, or none of the above, and in one sentence explain why: a) {(A,B), (B,C), (C,D), (A,C), (B,D), (A,D)} b) {(A,A), (B,A), (B,B), (C,C), (C,D), (D,D)} c) {(A,A), (B,A), (A,B), (B,B), (C,C), (A,C), (C,A), (D,D)} d) {(A,A), (B,A), (A,B), (B,B), (C,C), (D,D)} e) {(A,A), (B,A), (A,B), (B,B), (C,C), (C,D), (D,D)}
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}.
2. For each of these relations on the set {1, 2, 3, 4), decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4,4)} c) {(2, 4), (4, 2)} d) {(1, 2), (2, 3), (3, 4)} e) {(1, 1), (2, 2), (3, 3), (4,4)} f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
Let R be the following relation on the set of all people. R= {(a, b) : a and b share a common biological parent} Show whether or not R is an equivalence relation.
Let Z = {a, b, c, d}. What is the cardinality of the Power Set of Z, |P(Z)|?
What is an example of two sets with identical cardinality yet without one-to-one correspondence?
A car dealership sells cars that were made in 2015 through 2020. Let the cars for sale be the domain of a relation R where 2 cars are related if they were made in the same year. b) Describe the partition defined by the equivalence classes. How do I do this?
Let R be the relation on the set of all mathematicians that contains the ordered pair (a, b) if and only if a and b have written a published mathematical paper together. a) Describe the relation R[2] in the context of the problem.b) Describe the relation R* in the context of the problem.c) The Erdos number of a mathematician is 1 if this mathematician wrote a paper with the prolific Hungarian mathematician Paul Erdos. The Erdos number is 2 if a mathematician didn’t write a joint paper with Erdos but wrote a joint paper with someone who wrote a joint paper with Erdos, and so on (except that the Erdos number of Erdos himself is 0). Give a definition of the Erdos number is terms of paths in R. List a name of a mathematician with number: 1, 2, and 3
prove why each relation has or does not have theproperties: reflexive, symmetric, anti-symmetric, transitive. 1) Let A = { set of all people }, relation R: A x A where R = { (a,b) | a,b ∈ A, a is at least as tall as b }