1. Let Z be the set of integers and let m be a fixed positive integer. Define the relation by xy if and only if m divides x - y (equivalently ay is a multiple of m). This is called the relation congruence modulo m in Z. (a) Prove that this relation is an equivalence relation. (b) What are the distinct equivalence classes when m = 6? These are also known as the residue classes modulo 6 and the set of these residue classes is denoted by Z6.

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

A: Since you posted a multi sub part question but according to guidelines we will solve first three sub…

Q: Select the following relations on Z that are equivalence relations. O{(a,b)|a>b-1} {(a,b) | a² = 6²…

A:

Q: Consider the set K rational numbers) defined by x Ry if and only if E K, for x, y E Q+, is an…

A: K=2n:n∈ℤ A relation R is said to be an equivalence relation if it is reflexive, symmetric and…

Q: How do you find the equivalence class of 7 for mRn where R is a relation on the set of integers iff…

A:

Q: A relation R is defined on Z by xRy if and only if 8 divides 3x + 5y. Prove that R is an equivalence…

A: This is a question from relation.

Q: Define a relation on Z as follows: For m, n € Z, m~n ⇒ 5|(m - n). (5.1) Prove that is an equivalence…

A:

Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

A:

Q: 9. List the ordered pairs in the equivalence relation on A = {1, 2, 3, 4, 5} associated with these…

A: To find the ordered pair in the equivalence relation on A.

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

A:

Q: 1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint.…

A: Given that the F be a nonempty family of transitive relations.

Q: Let S be the equivalence relation on P({1,2,3,4}) — {0} defined by XSY if and only if max(X) – |X| =…

A:

Q: Prove relation B is an equivalence relation

A:

Q: 2. Suppose A = {1, 2, 3, 4, .. , 14, 15} and let R be the relation of congruence modulo 5 on the set…

A:

Q: Define a relation on Q by a b if a-b= for some k, n € Z and n > 0. (i). Prove that it is an…

A:

Q: Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3). (a) Prove that R…

A: (a) for any x in Z,       obviously xRx, as x2 = x2 mod 3       Hence, R is reflexive.        If…

Q: ) Let A = {1,2,3, 4} × {1,2, 3, 4}, and define a relation R on A by (1, Yı)R(x2, Y2) if r1 + y1 = x2…

A: Equivalence relation means it should satisfy reflexive, symmetric, Transitive conditions.…

Q: (a) Show that R is an equivalence relation. (b) List three members of each of the equivalence…

A: Equivalence relation: A relation R is called an equivalence relation in a set A, if it follows the…

Q: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

Q: 2 Let Z be the set of integers and let m be a fixed positive integer. Define the relation by y if…

A: To show that ~ is an equivalence relation we have to show it is reflexive symmetric and transitive.…

Q: 10. (1) A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

A:

Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

A:

Q: 1. In the R × R field explain why the definition of (x1, y1)~(x2, y2) if and only if x1 y2 = x2 y1…

A:

Q: vector spaces

A: We have to select the statements which are vector spaces. The set of all polynomials of degree…

Q: 6. Find all elements of the equivalence class of 12 under congruence modulo 5.

A: Given : 12 mod 5. To Find : All element of equivalence class of 12 under congruence modulo 5 . " As…

Q: ~R Let ~ be the relation on Z given by a ~R b if and only if there exists c E Z with gcd (c, 17) = 1…

A: The objective of this question is to prove that the given relation R is an equivalence relation and…

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 Rb if and only if a? = b° (mod 5).…

A:

Q: 2. Define a relation on Nx N by (n, m) (p, q) Is an equivalence relation? Why? if n+q=p+m.

A: This question is related to equivalence relation.

Q: Suppose that we are interested to count the number of integers not divisible by 7, 8, 9 between 1 to…

A: To find the integers between 1 to 10000 that are not divisible by 7,8 and 9. Firstly, find the L.C.M…

Q: 4. Let R be the relation on Z defined by xRy if x² = y² (mod 5). Prove that R is an equivalence…

A:

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…