Determine whether or not the following is an equivalence relation on N. If it is not,...

Determine whether or not the following is an equivalence relation on N. If it is not, determine which property fails. If it is, prove it.

(a) x ∼ y if and only if x|y (x evenly divides y).

(b) x ∼ y if and only if the least prime dividing x is the same as the least prime dividing y. In this case, instead if using N, use the set A = {2,3,4,5,...}.

(d) x∼yifandonlyifx=4.

2. Prove that x ∼ y if and only if sin x = sin y is an equivalence relation on R. What is the equivalence class of [π]?

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising...

Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation. (c) (x1,y1)R(x2,y2) in R×R if x1∗y2 = x2∗y1.

(1) In each part, decide whether or not the described relation is an equivalence relation. Explain...

(1) In each part, decide whether or not the described relation is an equivalence relation. Explain your answers. (a) A is the set of all residents of the United States. The relation ∼ on A given by: x ∼ y if x, y are in the same state at noon on November 3rd. (b) U is the set of all undergraduates at the University of Oregon. The relation ∼ on U given by: x ∼ y if x, y are...

Prove that the equivalence classes of an equivalence relation form a partition of the domain of...

Prove that the equivalence classes of an equivalence relation form a partition of the domain of the relation. Namely, suppose ? be an equivalence relation on a set ? and define the equivalence class of an element ?∈? to be [?]?:={?∈?|???}. That is [?]?=?(?). Divide your proof into the following three peices: Prove that every partition block is nonempty and every element ? is in some block. Prove that if [?]?∩[?]?≠∅, then ???. Conclude that the sets [?]? for ?∈?...

The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod...

The equivalence relation on Z given by (?, ?) ∈ ? iff ? ≡ ? mod ? is an equivalence relation for an integer ? ≥ 2. a) What are the equivalence classes for R given a fixed integer ? ≥ 2? b) We denote the set of equivalence classes you found in (a) by Z_5. Even though elements of Z_5 are sets, it turns out that we can define addition and multiplication in the expected ways: [?] + [?]...

Determine whether the following statements are correct (y) or not (n): and explain why.. ( )...

Determine whether the following statements are correct (y) or not (n): and explain why.. ( ) A system have a higher second law efficiency than the first law efficiency during a process ( ) It is impossible that the energy of one isolated system increases ( ) Adding moisture is the ONLY way to obtain saturated air from unsaturated air ( ) A Piston Cylinder device contains air. During a reversible, isothermal process the entropy of the air will never...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R...

Question 1. Equivalence Relation 1 Define a relation R on by iff . Prove that R is an equivalence relation, that is, prove that it is reflexive, symmetric, and transitive. Determine the equivalence classes of this relation. What members are in the class [2]? How many members do the equivalence classes have? Do they all have the same number of members? How many equivalence classes are there? Question 2. Equivalence Relation 2 Consider the relation from last week defined as:...

Let A = Σ*, and let R be the relation "shorter than." Determine whether or not...

Let A = Σ*, and let R be the relation "shorter than." Determine whether or not the given relation R, on the set A, is reflexive, symmetric, antisymmetric, or transitive.

A doctor wanted to determine whether there is a relation between a male's age and his...

A doctor wanted to determine whether there is a relation between a male's age and his HDL (so-called good) cholesterol. The doctor randomly selected 17 of his patients and determined their HDL cholesterol. The data obtained by the doctor is the in the data table below. Complete parts (a) through (f) below. Age, x HDL Cholesterol, y Age, x HDL Cholesterol, y 38 56 39 46 44 56 65 64 45 32 28 52 33 57 53 37 53 37...

A doctor wanted to determine whether there is a relation between a male's age and his...

A doctor wanted to determine whether there is a relation between a male's age and his HDL (so-called good) cholesterol. The doctor randomly selected 17 of his patients and determined their HDL cholesterol. The data obtained by the doctor is the in the data table below. Age_-_x HDL Cholesterol_-_y 36 58 44 56 46 33 30 55 57 36 52 42 62 42 60 39 25 48 38 44 68 60 28 52 52 34 27 47 53 39 47 ...

A psychologist would like to determine whether there is a relation between depression and aging. It...

A psychologist would like to determine whether there is a relation between depression and aging. It is known that the general population averages μ = 40 on a standardized depression test. The psychologist obtains a sample of n = 9 individuals who are all more than 70 years old. The depression scores for this sample are as follows. 50, 47, 41, 49, 44, 42, 43, 47, 48 On the basis of this sample, can the psychologist conclude that depression for...

Subjects