Let - be the relation on R\{0} given by x ~ y if and only if xy is positive. Prove that this is an equivalence relation.

Q: Let R be a relation that is symmetric and antisymmetric. Show that R is transitive

A: Let R be the relation which is symmetric and Anti-symmetric.We will show that R is transitive, that…

Q: A relation R defined on the set of real numbers is such that xRy and yRx = x = y: R is said to be A…

A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

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

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

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Q: Consider the following relation R on R: (x, y) ER if and only if y E Q. What are the properties of…

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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…

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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: Let R be a relation on the set A of ordered pairs of positive integersdefined by (x, y) R (u, v) if…

A: Given:

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: 3. Let R be the relation defined on Z by aRb if and only if 5 | (4a + b). Prove that R is an…

A: We have given relation on ℤ and defined by  aRb if and only if 5|(4a+b). We can  see here for any…

Q: Let R be the relation defined by x R y =x<y° Determine if R is an order relation on N? on Z? Justify…

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Q: Consider the equivalence relation on R defined by r ~ y if x – y € Z. Describe the quotient space…

A: In topology and related areas of mathematics, the quotient space of a topological space under a…

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…

Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Let R be the relation defined by x R y x² < y² Determine if R is an order relation on N? on Z?…

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Q: Prove that "is similar to" is an equivalence relation on Mnxn(F ).

A: We have to prove that “is similar to” is an equivalence relation on Mnxn(F ) i) Every matrix A is…

Q: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: Define a relation on Z by x ~ y if and only if x + 2y is divisible by 3. Show that this relation is…

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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 H be the set of all human beings. Consider the relation R defined on H by xRy if and only if x…

A: R defined on H by xRy if and only if x and y have the same biological mother. reflexivity : x &amp;…

Q: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.

A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…

Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

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Q: Let A = {1, 2, 3, 4} and R a relation on A whose matri: 1 0 1 0 0 1 0 1 is Mr = 0 0 1 1 0 0 1…

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Q: Let p be an integral domain and define a relation - on DxD such that bz0 and dz0 by (a,b) -~(c,d) if…

A: R is the symbol for relation.     Symmetric:    if (a,b)R(c,d) ∈ D×D   ⇒ad=bc   ⇒bc=ad   ⇒(c,d) R…

Q: Define a relation C from R to R as follows: For any (x, y) ∈ R x R, (x, y) ∈ C means that x² + y² =…

A: (x,y) ∈ R × R (x,y) ∈ C means    x2 + y2 =4

Q: Let f: X→Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Let E be…

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Q: Define a relation R on Z by xRy → x = y + 5k for k E Z. Prove that R is an equivalence relation by…

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

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Q: Let R be the relation on the set of real numbers such that x R y if and only if x and y are real…

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

A: The relation R is defined as xRy on natural numbers Nsuch that x and y share a common factor other…

Q: . Define a relation on R²\{(0,0)} by letting (x₁, y₁)~(x₂,Y₂) if there exists a nonzero real number…

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Q: Let R be a relation from X to Y and S be its reverse relation from Y to X. Prove or disprove that R…

A: According to the given information it is required to prove/disprove that R is injective and well…

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: 4|(x+3y)

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

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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 =…

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Q: RUR and RNR symmetric relations on X. are Show that R=Z xZ is an equivalence relation.

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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 =…

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Q: Give examples of relations R and S on Z such that  Both R and S are not equivalence relations but R…

A: We need to find two set which are on equivalence on Z but their sum do.  LEet, R= a&lt;b S=a&gt;b…

Q: Show that the relation R on Z defined by a R b if and only if 5a − 3b is even, for a, b ∈ Z, is an…

A: Given information: The relation R on Z defined by a relation b if and only if 5a − 3b is even, for…

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…

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Q: Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’ and y-y’ are integers. Prove…

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Q: Let - be a relation on N defined as: a ~b if a < (b – 1). (i) Is ~ reflexive? Prove your answer.…

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Q: Let S be the following relation on C: S = {(x, y) = C²:y-x is real}. Prove that S is an equivalence…

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Q: 18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence…

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Q: Let R be a relation on Z. Then R is an equivalence relation if it is defined by xRy if and only if O…

A: A relation R between the elements of a set is said to be equivalence relation if it is: Reflexive…

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…