The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and (x2, y2) in R^2 a) Prove that R is an equivalence relation. Be sure to use the correct format, including labelling b) ] Find the equivalence class [(2, 3)] and sketch it in the x-y plane.

Q: True or False? 11. If Γ = {[0, 1/n) | n ∈ N} then ∩Γ = [0, 1). 12. Let R be a relation on a set A.…

A: 11) if Γ ={[0,1n) | n∈N}  then ∩Γ=[0,1)         Γ= {[0,1), [0,1/2), [0,1/3), [0,1/4), [0,1/5),…

Q: Consider the collection of sets {An: n E R}, where for each n E R An := {(x, y) E R²: 2x + n}. y =…

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Q: Define relation R on (Z+ U {0}) × Z+ by (a, b)R(c,d) if ad = bc. Describe the equivalence class [(2,…

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Q: Let ∼be the relation on Z defined as follows: For every m, n ∈Z, m ∼n if, and only if, 5 | (m2 −n2…

A: (a) For every m ∈ Z, 5 | (m2 −m2) = 0      Hence, m ~ m for all m ∈ Z      Hence, ~ is reflexive.

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

A: Given: R is the relation defined on P1,....,100 by A R B.           A∩B is even.We need to check…

Q: Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find…

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Q: Theorem: If R is a fuzzy relation on the set X, then (R-1)-1 = R

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Q: ~ Define an equivalence relation on Z by a b if a - b is even. Show that . equivalence relation.…

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Q: 6. Are the following relations on R equivalence relations? Justify your answer! a) R = {(x, y): x -…

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Q: - b if and only if f (a) = f(b). Show that ~ is an Let f: A → B be a surjection. Define a ~…

A: Prove ~ is an equivalence relation. 1) Reflexive Since f is a function, f(a) = f(a) and so a ~ a for…

Q: Please help me out

A: Equivalence Classes: The set S is divided into equivalence classes by ~, which is an equivalence…

Q: efine the relation ~ on the set N of positive integers by a~b if and only if  a = b(10^k) for some…

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Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

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Q: Prove relation B is an equivalence relation

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Q: 1) Let S be the equivalence relation on R defined by xSy iff x-y Q. Then a) [√3]-(-√3+r,req) b)…

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Q: ) For arbitrary m, n € N, let (m, n) denote the greatest common divisor of m and n. or example,…

A: A relation ~ on any set R is said to be equivalence iff i) R is reflexive i.e a~a for all a∈R ii)R…

Q: 4. Describe all Consider the relation R on R x R defined by (x1, y1) R(x2, y2) iff x₁ - 2y2 = x₂ -…

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Q: #1 Verify the logical equivalence with laws:

A: Given expression is,

Q: For the points (x1, y1) and (x2, y2) in the plane R?let (x1, y1)~ (x2,Y2) mean that xỉ + yỉ = x + y…

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Q: For every element r in A, there is an element y in B such that (x, y) is in F. Can you please…

A: Relation: The relation from a set A to a set B is a subset of Cartesian product A×B. Function: The…

Q: Exercise 1.2. Determine whether the following relations ~ is an equivalence relation: (1) nm in Z if…

A: Step 1:Step 2:Step 3: Step 4:

Q: 2- i) Let B {(x,y):x,y eZ and 3(x-y)}. Show that B is an equivalence relation. Find equivalance…

A: A relation is said to be equivalence relation of it is reflexive, symmetric and transitive.…

Q: Suppose 1 and ₂ are equivalence relations on a set A. Let ~ be defined by the condition that a~ b if…

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Q: 1) Let R be the relation on Z+ such that (p, q) R(r, s) if and only if p- s = q ·r. That is, the…

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Q: 5. Let~ be an equivalence relation on a set S. We define f: S→S/~ f(x) = [a], where a € S such that…

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Q: Def (Function): A binary relation RCX X Y is called a function from X to Y, written f:X →Y if -…

A: The concepts have been explained with examples.

Q: Does the follorin relations hip define y as a real furction of x Why," AN

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Q: Prove that these relations on the set of all functions from Z to Z are equiva- lence relations.…

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Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: I.A) Antisymmetric Relation * A relation R on a set A is called antisymmetric if vavb((a,b) ER^…

A: I. Here, we have to select the best answer for each item. I. A) Antisymmetric Relation. I. B)…