Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

Q: Prove that in a topological space X, if U is open and C is closed, then U –- C is open and C – U is…

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

A: Transitive means aRb and bRc then aRc

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: The set G = :a is closed under addition %3D 0 2a

A: Use the closure property of addition.

Q: Prove that the set {a E R: a2 not equal to 6} is open.

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Q: The relation R defined on a set A is antisymmetric if (a, b) ER = (b, a) E R for (A) Every (a, b) ER…

A: A relation R is said to be antisymmetric if (a,b)∈R and (b,a)∈R implies a=b

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

A: Given:  R is a relation defined on P 1, 2, … , 100 by A R B  if and only if A ∪ B is even. We have…

Q: 1. if R and T are anti-symmetric relations on a set A then (R U T) is anti-symmetric relation on

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Q: Prove the understated statements concerning the relation < on the set Z of all integers . If    x…

A: x = y  and     O < z

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 Z = {0, 1, 2, 3} and the relations R, S, and T on Z are as follows:

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Q: Let (X, I) be a topological space and suppose that A and B are subsets of X such that ACB. Prove…

A: Let x ∈A°. Then by definition, there exists an open set U  such that  x∈U⊆A.Since A⊆B hence x∈U⊆B ,…

Q: If R is a symmetric relation on a set A then a. None O b. mij = 0 for alli %3D O C. mji = 0 for alli…

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Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: Let R be a relation defined on P(Z) defined by (A, B) E R if and only if AN B + Ø. Prove or disprove…

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Q: Show that given relation is Symmetric For X, y e R say x is congruent to y modulo Z if x - y is an…

A: This question is related to number theory.

Q: let R be the relation on the set A = {1, 2, 3, 4, 5, 6} defined by a R bif and only if a = 2b.

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Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

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Q: 1. Prove or Disprove: For any set A, there exists a relation R on A such that R is both symmetric…

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Q: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…

A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric)  So, R∘S=S∘R

Q: Which of the following is/are true for the relation R, where a is related to b (aRb), if and only if…

A: Suppose, A be any set and suppose R:A->A is a relation. Then, R is reflexive: if for every x in…

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: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…

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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: Let R be the relation on the set of people with doctorates such that (a, b) ER if and only if a was…

A: Let A be the set of all people with doctorates and R be a relations on A defined by a,b∈R if and…

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 the relation R defined on the set of integers ℤ by (a,b) Î R if a – b = 6k for some k Î ℤ…

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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: Suppose f, g, and h are all mappings of a set A into itself. a. Prove that if g is onto andf 0 g = h…

A: Given that, f,g and h are all mappings of a set A into itself. If g is onto and f∘g=h∘g  To prove…

Q: A relation \(R\) on a set \(AV) is called anti-symmetric if O (a.b) \in R\) implies ( (b,a) \in R\)…

A: Anti symmetric relation is as follows: If (a, b) and (b,a) lies in R then must be a=b.

Q: (b). Let Y = {a, e, o, i, u}, take two relations R and S on Y and Prove that (SoR)-1 : R-1os-1.

A: composition of relations

Q: Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric and…

A: Let  A be any we define a relation R on A s.t

Q: Show that given relation is transitive For X, y € R say x is congruent to y modulo Z if x – y is an…

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Q: 1. Suppose R and S are symmetric relations on a set A. Prove that Ros is symmetric if Ros=SOR

A: Given:  To prove R and S are symmetric relations.

Q: 1. if R and T are anti-symmetric relations on a set A then (R U T) is anti-symmetric relation on А.

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Q: Let A, B, C be sets. Prove that if f: A → B and g: B → C are bijections, the composition gof: A → C…

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Q: For (a, b),(c,d) EN×N define the relation (a, b) ~ (b,c) if and only if ab = cd. Is this relation…

A: Explanation of the answer is as follows

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

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

Q: Give examples of relations R and S on Z such that  Both R and S are not equivalence relations but R…

A: We need to find two set which are on equivalence on Z but their sum do.  LEet, R= a<b S=a>b…

Q: Let (X7) be a topological Space and Show that A is open and closed m (xx) iff %3D

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Q: f A∩B≠∅ and A∩Bc≠∅ then show that A∩X\B̅≠∅ and A∩B̊≠∅

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Q: Let A={(m,n)∣m,n∈N2 and gcd(m,n)=1}, and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc.…

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Q: Determine which of the folowing relations R on Z are oquivaknee relatinns. If R is not an equivakner…

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Q: Let G={(m,n)∣m,n∈N x N and gcd(m,n)=1}, and define the relation q on G according to (a,b) q…

A: A totally ordered set is a set G together with a relation ≤ that satisfies the following conditions.…

Q: 5. Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric…

A: Symmetric : If aRb , then bRa , for every a , b∈A Antisymmetric : If aRb and bRa , then a=b

Q: Let R and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

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Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…

A: The complete solutions is in given below

Q: Let S, T, and R be sets. Assume that f: S-» T, and g: T -» R are maps Assume that gf is onto. Does f…

A: Assume that gf: S→R is onto,Then, by the definition of onto:

Q: Prove the understated statements concerning the relation < on the set Z of all integers .   If O…

A: Prove the understated statements concerning the relation < on the set Z of all integers .   If O…