Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by

(i) R = {(a, b) : |a – b| is a multiple of 4}

(ii) R = {(a, b) : a = b}

is an equivalence relation. Find the set of all elements related to 1 in each case.

### Want to see the full answer?

Check out a sample Q&A here*star_border*

## Related Algebra Q&A

Find answers to questions asked by students like you.

Q: To find an equivalence class of x ϵ X determined by the relation R ⊆ X ⨯ X, R must be an equivalence…

A: Equivalence Relation: A relation R in X is called an equivalence relation if it is reflexive,…

Q: Let R be the relation consisting of all pairs (x, y) such that x and y are strings of uppercase and…

A: Let R be a relation on a set A R is an equivalence relation if it is reflexive, symmetric,…

Q: Show that the relation R is an equivalence relation in the set A = { 1, 2, 3, 4, 5 } given by the…

A: Given- Set A={1,2,3,,4,5}. To Show- That the relation R is an equivalence relation in the set A = {…

Q: 1. For (a, b) and (c, d) in R², define (a, b)R(c, d) by a² – c² = b2 - d². Determine whether the…

A: The given problem is to prove that the relation is equivalence relation, to be equivalence it must…

Q: For each relation on {0, 1, 2, 3} below show that the relation is an equivalence relation, or show…

A: I have used the definition of equivalence relation

Q: Let A = Z and B = Z\{0}. Show that the relation R on A x B defined by (a, b)R(c, d) if ad = bc, is…

A: Click to see the answer

Q: If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence…

A: Given, R1 and R2 are equivalence relations in a set A. So, both R1 and R2 are Reflexive, Symmetric…

Q: 2. A relation p is defined on the set of all real numbers R by xpy if and only if |x- yl < y, for x,…

A: Click to see the answer

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

A: Click to see the answer

*question_answer*

Q: A relation on a set A is circular iff , y, z x~y and y~z implies z~x. Prove that a relation is an…

A: The relation on a set is a subset of the Cartesian product of the set. Therefore, each element is in…

*question_answer*

Q: Let R be the relation on the set of all people who have visited a particular Web page such that x Ry…

A: Given

*question_answer*

Q: Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b} is an…

A: Given, the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b}.

*question_answer*

Q: Give an example of two equivalence relations R and S on

A: Consider that R and S are equivalence relations on A=1, 2, 3.

*question_answer*

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

A: Click to see the answer

*question_answer*

Q: Either show the following relations on {0, 1, 2, 3} is an equivalence relation, or show which…

A: Either show the following relations on {0, 1, 2, 3} is an equivalence relation,

*question_answer*

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

A: Thanks for the question :)And your upvote will be really appreciable ;)

*question_answer*

Q: Let R be the relation on A × A, where A = {1,2,3,4}, that is, ele- ments of this relation are pairs…

A: Click to see the answer

*question_answer*

Q: binary relation R is defined on the set of real numbers by xRy ⇔ lx - yl ≤ 1. Determine whether or…

A: R is not an equivalence relation. Because, R is not transitive. The Counter example is as follows…

*question_answer*

Q: Let A be the set of all students of a boys school. Show that the relation Rin A given by R = {(a, b)…

A: It is given that, A is the set of all students of a boys school.Show that R is an empty…

*question_answer*

Q: Let A = {0, 1, 3, 5, 7, 8} and R be a relation defined on A such that R= {(x , y)| 4 divdes (x –…

A: We are given the set A = {0, 1, 3, 5, 7, 8} and the relation R defined on A such that R = {(x, y)| 4…

*question_answer*

Q: Suppose R is a relation defined on the set A = {1,2, 3, 4}. If R is an equivalence relation, what is…

A: Given below the detailed solution

*question_answer*

Q: Prove that the relation of congruence is an equivalence relation.

A: Click to see the answer

*question_answer*

Q: Prove that the relation of congruence is an equivalence relation.

A: A relation is said to be an equivalence relation if and only if the relation is reflexive, symmetric…

*question_answer*

Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

A: Click to see the answer

*question_answer*

Q: The relation defined as x ~ yx ≡ y (mod7) for x, y ∈ Z in Z Write the equivalence classes by writing…

A: Relation R is defined in Z as x≡y(mod 7) (difference of x and y is multiple of 7) Reflexivity : For…

*question_answer*

Q: Let R = {(f, g) | f(1) = g(1)} be a relation defined on the set of func- tions from Z to Z. Is R an…

A: Let R be a relation on a set A. Then R is said to be an equivalence relation if it is reflexive,…

*question_answer*

Q: Let S be a set of distinct symbols. Show that the relation defined onW(S) in this chapter is an…

A: Click to see the answer

*question_answer*

Q: (a) Ri (b) R2= {(1,2), (1,3),(1,4),(2,2),(3,3),(5,3),(5,4),(5,2)} /hich of them is an equivalence…

A: Any relation is equivalence relation if satisfy the following conditions where is set of elements a)…

*question_answer*

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

A: A ={ (a,b) : where a and b are positive integers} R= {(a,b) and (c,d) : a+d= b+c } We have to…

*question_answer*

Q: Given the relation R- ((1,1), (2.2) (3.31(4.4). (1,2) (2.1). (3,4) (4.3)) on the set X- (1,2.3.4),…

A: Given a relation R=1,1, 2,2, 3,3, 4,4, 1,2, 2,1, 3,4, 4,3 on a set X=1,2,3,4. Clearly the relation…

*question_answer*

Q: Let R be the relation on the all integers given by xRy iff x- y 2k for some integer k. (1) Prove…

A: Click to see the answer

*question_answer*

Q: Prove and say the name of the following equivalence relation x ~ y ⟺ x - y = 4k, k ∈ Z

A: x ~ y iff x - y = 4k , k in Z Implies that x is congurent to y mod 4.

*question_answer*

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…

A: Click to see the answer

*question_answer*

Q: Obtain the 16 relations of the set {0,1}, indicate which of these are reflexive, symmetric and / or…

A: Click to see the answer

*question_answer*

Q: 9.15. A relation R is defined on Z by a Rbif |a – b| < 2. Which of the properties reflexive,…

A: Given : A relation is R is defined on Z is given by a R b if a-b≤2 Which of the properties…

*question_answer*

Q: let R be an equivalence relation and let [x] be the equivalence class of x, prove x belong to [x]

A: Click to see the answer

*question_answer*

Q: Given the relation R- {(1,1), (2.2). (3,3).(4.4), (1,2). (2,1). (3,4), (4,3)) on the set X…

A: Click to see the answer

*question_answer*

Q: A relation that assigns to each element x from a set of inputs, or ________, exactly one element y…

A: Domain→The set of all possible values which qualify as inputs to afunction is known as the domain of…

*question_answer*

Q: Let S be a nonempty set. Consider the relation ~ on P(S) defined byA ~ B if A U BC = S. Is the…

A: Let S be a nonempty set. Consider the relation ~ on P(S) defined by A~B if A∪Bc=S To show: "~" is…

*question_answer*

Q: Show that R is an equivalence relation.

A: Let R be the relation consisting of all pairs (x, y) such that x and y are strings of uppercase and…

*question_answer*

Q: Let S be the set of real numbers. If a,be S, define a ~ b if a -b is an integer. Show that is an…

A: Let S be the set of real numbers. Given the relation R is define a~b if a-b is an integer. To show…

*question_answer*

Q: 6. Consider the relation R = {(a, b), (a,c), (c, b), (b,c)} on the set A = {a, b,c}. Which of the…

A: Click to see the answer

*question_answer*

Q: 6. Consider the relation R = {(a, b), (a,c), (c, b), (b,c)} on the set A = {a, b,c}. Which of the…

A: Reflexive Relation -A relation is a reflexive relation if every element of set A maps to itself, i.e…

*question_answer*

Q: 6. Consider the relation R = {(a, b), (a,c), (c, b), (b,c)} on the set A = {a, b,c}. Which of the…

A: Given set is A={a,b,c}

*question_answer*

Q: Let A = {3, 4, 5} and B = {1, 2, 3, 4} and define a relation R: A → B as follows: For all (x, y) ∈ A…

A: A=3,4,5B=1,2,3,4R=x,y: x∈A, y∈B ∋xy>7 a. R=3,3, 3,4, 3,5, 4,2, 4,3, 4,5, 5,2, 5,3, 5,4, 5,5

*question_answer*

Q: If R = {(1,2),(2,1),(2,2),(3,4)} is a relation on the set {1,2,3,4}.The reflexive closure of R is :

A: Given that R=1,2,2,1,2,2,3,4 is a relation on a set 1,2,3,4 Consider A=1,2,3,4

*question_answer*

Q: 5. Let A=Z and R, ={(x,y):x-y is divisible by 5} be a relation on A . That is x R,y if and only if…

A: Click to see the answer

*question_answer*

Q: Let A 1, 2} and B = {1, 2, 3} and define a relation R from A to B as follows: Given any (x, y) E A x…

A: Given

*question_answer*

Q: Determine whether the following relations are equivalence relations. For those relations that are…

A: Click to see the answer

*question_answer*

Q: Determine whether the following relations are equivalence relations. For those relations that are…

A: Consider the given relation is given as: