Let R be a relation on the set A = {x € Z:1 < x < 100} defined as follows: x R y if and only if x-y <1. Is R equivalence relation? Fully justify your answer.

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…

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Q: 4. Let f : X → Y define a relation on the set X by „Ry if and only if (x) = fy). Is f an equivalence…

A: f is an equivalence relation if f satisfies three properties. 1. if f is a reflexive relation  2. if…

Q: Suppose that R is a reflexive and transitive relation on a set A. Define a new relation E on A by…

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Q: Let A={a,b,c,d,e}. Find a relation R on A which is symmetric and transitive but not reflexive

A: Reflexive relation means every term of a set is related with itself

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…

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Q: PROVE THAT IF A RELATION X ON NON EMPTY SET B IS REFLEXIVE AND TRANSITIVE THEN THE RELATION X…

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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: Let S be the relation on the set R^2 defined by (x, y) S (w, z) if |x| + |y| = |w| + |z|. Prove or…

A: Definition of an Equivalence Relations: A relation R on a set A is an equivalence relation if…

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: 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: (i) Let A = {1,2,3,4,5), then either find a relation on A that is not reflexive on A and not…

A: Solution:

Q: S is an equivalence relation. 2. Define the relation R on Z as follows: for x, y E Z, rRy if and…

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Q: Define the relation R on Z as follows: for x, y ∈ Z, xRy if and only y−x ≥ 0. Determine whether R is…

A: partial order means reflexive and antisymmetric.

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

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Q: Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists CEP…

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Q: (b) Let - be the relation on C defined by z ~ w if and only if z + w e R. Proye that is an…

A: This can be solved as follows:

Q: Let R be a relation from A={−1,0,1,2} to B={a,n,i} such that (x,y) provided that x is non-negative…

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Q: Let ≥ be a relation defined on sets A and B by A ≥ B iff there exists a surjective function f : A →…

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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…

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

A: R is an equivalent relation iff R is reflexive. R is symmetric. R is transitive.

Q: A relation R defined on the given set A will satisfy

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Q: Let A be a set and let ∼ be an equivalence relation on A. Let x, y ∈ A. Prove that [x] = [y] if and…

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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: 4. Let R be the relation xRy if 3| (x – y) on Z. Prove that R is an equivalence relation.

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Q: 2. Let R be a relation defined on the set of integers Z where a R b if a +b is even. Determine…

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Q: Define a relation Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x R,…

A: The relation Q on the set R×R is defined as follows. For all ordered pairs w, x and y, z in R×R, w,…

Q: Let A be a nonempty set. Let ~ be a relation on ℘(A)defined by: Letting B and C be arbitrary…

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Q: Define a relation C from R to R as follows: For any (x,y) R x R, (x,y)C meaning that x2 + y2 = 1…

A: Given

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 S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an…

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

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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: 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,…

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

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Q: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

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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: Let S be the following relation on C: S = {(x, y) = C²:y-x=bi for some real number b}. Prove that S…

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Q: Let A = Z. Define R to be the relation where a~b if ab > 0. Then R is equivalence relation.

A: We have given the set A = { Z : the set of integers }  R= { (a,b) : a,b belongs to Z and a.b &gt;0}…

Q: Let R be a relation defined on Z by xRy if and only if x-y=7k for kez. The equivalence class of [5]…

A: Given relation is xRy iff x-y=7k So, equivalence class of 5 will contain ....-9, -2, 5, 12, 19,....…

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

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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 the relation on N defined by a R b if either a | 2b or b | 2a. Is R an equivalence…

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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…

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