Prove that the given relation is an equivalence relation, and describe the distinct equivalence classes of this relation. A is the set of all students at your college. S is the relation defined on A as follows: For every x, y E A, x S y → x is the same age as y.

Q: Fix a set X, and let S be a collection of equivalence relations on X. Is s U an equivalence relation…

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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: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: Let = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence…

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Q: Given a non empty set X, consider P(X) which is the set of all subsets of X.      Define the…

A: The relation is defined as follows.

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

A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…

Q: 1. Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible…

A: According to question given that Set Z in which the relation R is defined by aRbIf and only if a+3b…

Q: A relationR on a set A is called equivalence if R is:

A: Equivalence Relation: A relation R on a set A is said to be an equivalence relation if and only if…

Q: Let S be the equivalence relation on {− 1, 0, 1} × {0, 1, 2, 3} defined by (a, b )S(c, d) if and…

A: Let A  = {-1,0,1}×{0,1,2,3}  i.e A=…

Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

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Q: (b) Let R be the equivalence relation on the set S = {1,3, 5, 7, ..., 49} defined by x Ry if and…

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Q: Define the relation ~ on the set Z of integers by m~n if and only if m^2 = n. Show that ~ is not…

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Q: on P(X), where X = Y, Z E P(X), we say Y - Z if and only if Y has the same number of ele- Define a…

A: Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say  Y~Z if and only…

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: Define the relation on the set Z of integers by if and only if a ~b if and only if a = b + 5k for…

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Q: for an equivalence relation R on A, both sets, the class of equivalence classes of R —A/R— is a set.

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Q: Let the relation non z defined by x ny if and only if 3/(x+2y) Show that n is an equivalence…

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Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…

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Q: F is the relation defined on Z as follows: Vr, n E Z, mFn =7|(m² – n²). (a). Prove that F is an…

A: Equivalence relation and classes

Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})2: y/x = 2k for some integer k}.…

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Q: A particular relation * is defined on the set Z-{0} by x * y if and only if every prime factor of x…

A: Given that a particular relation is defined on the set ℤ-0 by…

Q: Subject: Set Theory

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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: Given the relation R- {(1,1), (2.2). (3,3).(4.4), (1,2). (2,1). (3,4), (4,3)) on the set X…

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Q: Let S' = {x € R² : ||x|| = 1} C X. and endow S' with the same equivalence (b) relation - as above.…

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Q: 1. For (a, b) and (c, d) in R², define (a, b)R(c, d) by a² – c² = b2 - d². Determine whether the…

A: The given problem is to prove that the relation is equivalence relation, to be equivalence it must…

Q: For each of the following sets A and binary relations ∼, decide whether ∼ defines an equivalence…

A: (a)

Q: 1. Consider the relation R on the set A = {0,1, 2, 3, 4}, defined by: %3D aRb + a = bc and b= ad,…

A: The given set is A=0,1,2,3,4 and the relation is defined as aRb⇔a=bc and b=ad (a) Check whether R an…

Q: 3. Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: We are given  that R is a transitive relation on a set A  Transitive Relation  A relation on a set X…

Q: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

Q: Define a relation on Z by a R b if a² = b². a. Prove that R is an equivalence relation. b. Describe…

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Q: A relation R is called atransitive if aRb and bRc implies cRa. Show that R is reflexive and…

A: Given a relation R is called atransitive if aRb and bRc⇒cRa Now shows in the following that R is…

Q: et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈…

A: This is the problem of Relation.

Q: Define a relation R on the set of all integers Z as follows: for all integers m and n, m R n → m = n…

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Q: ) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: Problem 17. (a) Prove that <r is a reflexive, transitive relation on P(E*). (b) Prove that =r 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: Determine, with justification, whether the following relations are equivalence relations. If so,…

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Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in  R…

Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

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Q: A relation on a set A is circular iff , y, z x~y and y~z implies z~x. Prove that a relation is an…

A: The relation on a set is a subset of the Cartesian product of the set. Therefore, each element is in…

Q: Which of the following definitions for the relation R (defined below) on A would constitute an…

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Q: Let A be a nonempty set and let P be a partition of A. Define a relation R (corresponding to P) on A…

A: Given, A=1,2,3 We can write the above set as, 1,2,3×1,2,3=1,1,,1,2,1,3,2,1,2,2,2,3,3,1,3,2,3,3

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: 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 and x be relations on Z defined as follows: For a, be Z, a~b if and only if 2 divides a +b. •…

A: According to the given information, it is required to solve the question 10.

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: (5) Let A={-2,-1,0,1,2}. R is an equivalence relation defined as: for all x,y E A, xRy 2| (x-y?).…

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Q: Show that the relation R defined on the set of integers ℤ by (a,b) ∈ R if a – b = 6k for some k ∈ ℤ…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.