Let A be a nonempty set and R be a relation on A such that domain(R) = A. Prove that if R is symmetric and transitive then R is an equivalence.

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A: A relation R is said to be an equivalence relation, if it is  Reflexive : For every a∈R→(a,a)∈R…

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A: Equivalence relation

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A: The set A×A={(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} R={(1,1), (3,1), (2,1),…

Q: 1. Define a relation R on R? by stating that (a, b)R(c, d) if and only if a? + b <c? +d². Show that…

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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 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,

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A: A relation R on ℤ is defined by: a R b if and only if 5 | a2-b2 . We have to prove that R is an…

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Q: Let {A1, A2, . . .} be a partition of a set X, and R the relation { (x, y) ∈ X 2 : there exists j…

A: An equivalence relation is a binary relation between two elements of a set that satisfies three…

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A: Given problem is :

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A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (a, b, c, d) and define a relation R on A as follows: R = ((a, a), (b, b), (a, b), (c, c), (b, a),…

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Q: Let R be a relation on a nonempty set A such that IA CR, R = R¹ and RoRCR. Prove explicitly that R…

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Q: Let A = {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3) and define a relation R on A as follows: For all x, y…

A: Given : A = {-6, -5, -4, -3, -2, -1, 0, 1, 2, 3} and R is an equivalence relation on A defines as ;…

Q: (b) Let {A₁, A2,...} be a partition of a set X, and R the relation {(x, y) = X²: there exists j such…

A: An equivalence relation is a binary relation between two elements of a set that satisfies three…

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Q: 17. Prove that if R is a symmetric, transitive relation on A and the domain of R is A, then R is…

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Q: Let L be a relation on R such that for all x and y in R, x L y if and only if x < y. Give a counter…

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Q: Sketch a digraph for a relation on the set {a, b, c, d} that is irreflexive and symmetricbut not…

A: Denote R by the relation on the set S={a, b, c, d} that is irreflexive, symmetric and not…