Define a relation R on Z by xRy → x = y + 5k for k E Z. Prove that R is an equivalence relation by using direct proofs for the reflexive, symmetric and transitive properties.

Q: Let A C B and let S be an equivalence relation on B. Prove that R = {(x,y) E A × A : (x, y) E S} %3D…

A:

Q: Let A = {x, y, z}. Write down a relation R on A that is not reflexive, not sym- metric, and not…

A:

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

A:

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. Is R reflexive?…

A: Given:  R is a relation defined on P 1, 2, … , 100 by A R B  if and only if A ∪ B is even. We have…

Q: Consider the relation K on N defined by Knm iff n <m.Which statement below is true? K is an…

A: It is Irreflexive.

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

A:

Q: Show that “is unitarily equivalent to” is an equivalence relation on Mn×n(C).

A:

Q: Let Z be the set of integers. Consider R = {(r, y) e Z x Z: 7 divides x-y} CZ x Z. Show that R is an…

A:

Q: Let R be a relation on Z defined by (x,y)ER if and only if 5(x-y)=0. Formall state what it means for…

A:

Q: 3. Let R be the relation on the set of Z as xRy as xRy if and only if x 2 + y2 is even. Is this R…

A:

Q: Consider a relation R on Z- {0} defined by the rule that (x, y) E R if and only if xy > 0. a) Prove…

A: Given the relation R on ℤ-0 defined by         x,y∈R if and only if xy&gt;0.

Q: Let f: X → Y and define x~y if f (x) = f (y). Show that - is an equivalence relation on X.

A: The objective is to show that ~ is an equivalence relation on X.

Q: b. Define a relation SER XR by S ={ (x,y) E R × R |x – y € Z }. Prove that S is an equivalence…

A: By equivalence relation, we mean that the relation is reflexive, symmetric, and transitive.Here we…

Q: Show that the relation R in the set A of all the books in a library of a college,given by R = {(x,…

A:

Q: Let D  be an integral domain and define a relation ~ on DxD such that b is not equal to 0  and d is…

A:

Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

A:

Q: Let R be the relation on N defined by x Ry if x and y share a common factor other than 1. Determine…

A: The relation R is defined by xRy if x and y share a common factor. We need to check reflexivity and…

Q: Let R be a reflexive and transitive relation on A. Show that RN R¯' is an equivalence relation -1 on…

A: First let, A= set of integers. Then, let, R={(a,b):a,b in A} so,

Q: Consider the relation S on N defined by uSy + 5|(y + u). Determine whether S is symmetric, reflexive…

A:

Q: Let R = { ( a,b ) | a,b ∈ Q and a − b ∈ Z } . Prove that R is an equivalence relation on Q

A:

Q: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…

A:

Q: Give an example of a nonempty relation R on A = {1, 2, 3, 4} which is a. symmetric and transitive…

A:

Q: Let R be the relation on R defined by R= {(x, y) E R × R | x² + y² = 4 or x = y²}. Answer each of…

A: Given that R be the relation on ℝ defined by,R=x,y∈ℝ×ℝx2+y2=4 or x=y2

Q: Let S be the following relation on C\{0}: S = {(x, y) = (C\{0})² : y/x is real}. Prove that S is an…

A:

Q: Define a relation on Z by a R b if a² = b². a. Prove that R is an equivalence relation. b. Describe…

A:

Q: Consider a relation R on Z such that for x, y E Z, (x, y) E R if and only if xy = 0 (mod 11) Which…

A: Reflexive relation: A relation S on a set A is called reflexive if every element of A is related to…

Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…

A: Consider the given information.

Q: Prove whether the following statements are true or false: b) The relation R defined by xRy7[(x-y)…

A:

Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

A:

Q: 1. A relation R is defined on Z* × Z* by (m,n)R(p, q) → m+q = n + p . (a). Prove that R is an…

A:

Q: Let R be a relation on Z defined by R = {(p, q) E Z × Z |p – q is a multiple of 3}. (a) Show that R…

A: Since you have asked a question having multiple subparts, we will solve the first three subparts for…

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 =…

A:

Q: Define a relation T on R by xT y if and only if (sin^2) x + (cos^2) y = 1. (a) Prove that T is an…

A:

Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})² : y/x = 2k for some integer k}.…

A:

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. s R reflexive?…

A: We will check one by one what properties R satisfies and which not.

Q: 8. Let A = Z* U {0}, define relation R on A × Z* by (k,l)R(m,n) → kn = lm. (a). Prove that R is an…

A:

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 =…

A:

Q: 6. (a) Let ~ be the relation on Mat2(R) defined by: A ~ B if and only if the (2, 1) entry of A – B"…

A: This problem can be solved as follows:

Q: Let R be the relation defined on P({1,.…, 100}) by ... ARB if and only if |AUB| is even. Is R…

A: Relation: - A subset of the productA× B could be a relation R from a non-empty set A to a non-empty…

Q: In this exercise, you want to show that a relation RC A? is transitive + R" C R for all n EN=…

A:

Q: . Let R be the relation on Z→ Z such that ((a,b),(c,d)) eR = a+d =b+c. Show that R is an equivalence…

A: An equivalency relation on a set is a reflexive, symmetric, and transitive relationship for…

Q: Define a relation R on Z by declaring that xR y if and only if x^2 ≡ y^2 (mod 4). Prove that R is…

A:

Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…

A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…

Q: Let S be the following relation on C10). S=[(x,y) E (C\[0]): y/x is real). Prove that S is an…

A:

Q: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…

A: A relation to be an equivalence relation must satisfy the following three properties:  1. Reflexive…

Q: Define a relation on Z by setting xRy if xy is even. a. Give a counterexample to show that R is not…

A:

Q: For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence…

A:

Q: Let R be a relation and S its reverse. Show that R is injective if and only if S is well defined,…

A: Consider the provided question, Given that, R be a relation and S its reverse. Let R is a relation…

Q: Let ∼ be the relation on Mat2(R) defined by: A ∼ B if and only if the (2, 1) entry of A − BT equals…

A: