Let S be the set of real numbers. If a, b e S, define a - b if a – b is an integer. Show that ~ is an equivalence relation on S.

Q: Let S be the set of all real numbers and let R be a relation in S, defined by R = {(a, b): a < b?.…

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Q: Let A be the set of all students of a boys school. Show that the relation Rin A given by R = {(a, b)…

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Q: Let N be the set of all natural numbers and let R be a relation in N, defined by R = {(a, b): a is a…

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Q: Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a)…

A: 11. Let A be a nonempty set and ~be an equivalence relation on A. (b)  We know that for each a, b∈A…

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Q: Prove that the relation of congruence is an equivalence relation.

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Q: Let S be the set of real numbers. If a, b E S, define a ~ b if a - b isan integer. Show that ~ is an…

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Q: Let S be the set of integers. If a, b E S, define aRb if a + b is even.Prove that R is an…

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Q: Let S be the set of ordered pairs of positive integers, and define (a,b) R (c,d) to mean that 2a-d =…

A: A relation R is transitive on a set A if xRy and yRz then xRz for all x,y and z belong to A. Now,…

Q: Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b} is an…

A: Given, the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b}.

Q: Prove that the following relations are equivalence relation: On Z× (Z−{0}), with (a,b)∼(c,d) if…

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Q: Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈…

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Q: Let S be the set of all real numbers and let R be a relation in S, defined by R = {(a, b) : a < b³}.…

A: Given relation is R = {(a, b) : a ≤ b3}

Q: In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z x N.…

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Q: Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by(i) R = {(a, b) : |a –…

A: Given: Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by(i) R = {(a, b)…

Q: Determine whether the relation R on  the set of all people is reflexive,symmetric, antisymettric…

A: Consider the provide question,

Q: Let R be the relation over the set of all people where aRb iff a is the same age as b. Determine and…

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Q: Determine whether or not the relation R on the set of all people is reflexive, symmetric,…

A: We will find whether the relation is reflexive, symmetric or transitive as following.

Q: Consider the relation R = {(a, b), (a, c), (c, b), (b, c)} on the set A = {a, b, c}. Is R reflexive,…

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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: Let S be a set of distinct symbols. Show that the relation defined onW(S) in this chapter is an…

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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: Describe the equivalence classes of the relation R = {(a, b) : a E Z,beZ, 13 divides (a - b)}

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Q: Let ~ be defined on the set X = {0,1,2,3,4,5,6,7,8,9} by x ~ y + x2 = y? mod 5. Prove that is an…

A: Equivalence relation are those which are reflexive symmetric and transitive

Q: Let the relation R defined on the set of positive integers as (x, y) ER if 3 divides X - y Determine…

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Q: A relation R, is defined on Z, the set of integers by aRb + a - b is even. Show that R is…

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Q: let R be an equivalence relation and let [x] be the equivalence class of x, prove x belong to [x]

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Q: Let R be relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) e R…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: (d) Give an example of an equivalence relation on the set f1,2, 3} with exactly two equivalence…

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Q: Use a particular counterexample to explain why R fails to be an equivalence relation is the…

A: We take counter example   9R10,10R11⇒9R  11

Q: The relation defined as x ~ yx ≡ y (mod7) for x, y ∈ Z in Z Write the equivalence classes by writing…

A: Relation R is defined in Z as x≡y(mod 7)  (difference of x and y is multiple of 7) Reflexivity : For…

Q: Define: what is an equivalence relation?

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Q: Obtain the 16 relations of the set {0,1}, indicate which of these are reflexive, symmetric and / or…

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Q: let S be the set of real numbers. if a,b is an element of S define a~b if a-b is an integer,show…

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Q: Let A be a set and R be a reflexive relation on A. Prove that R is an equivalence relation if and…

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Q: Let the relation R on the set of ordered pairs of positive integers be defined as (a, b) R (c, d) a,…

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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…

Q: Determine whether or not the relation R on the set of all people is reflexive,  symmetric,…

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Q: . Let R be a relation defined on the set A = N x N where (a, b) R (c, d) if a³ s an equivalence…

A: Introduction: Any two elements are said to be equivalent if they have an equivalence relationship. '…

Q: Let R be the following relation on the set of all people. R= {(a, b) : a and b share a common…

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Q: For universe S = R², define relation ~= as: (a,b) ~= (c,d) < 2a - b = 2c - d Prove that ~= is an…

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Q: Let R be an equivalence relation on A. Find the maximum number of classes R can make if JA| = 80 and…

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Q: Let R be the relation on the set of all people who have visited a particular Web page such that x Ry…

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Q: Consider the relation R on the set Z defined by a R b if and only if 3|(a+2b) for a, b ∈ Z. Show…

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Q: Define a relation R on the integers Z saying that (m, n) is in R if m2 is equivalent to n2 (mod 7).…

A: First to prove the relation, R, on set of integers Z is equivalence relation. R is defined as:

Q: Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ER…

A: A ={ (a,b) : where a and b are positive integers}  R= {(a,b) and (c,d) : a+d= b+c }  We have to…

Q: 8. Let A be a set of nonzero integers and let R be a relation on A × A defined by (a, b)R(c, d)…

A: Equivalence relation