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.

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

A:

Q: Define the following infinite collection of subsets of the positive integers: A1 = {x|x E Z* and 0 <…

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

Q: Is the following statement true or false? If R is any symmetric relation on a set A, then R is…

A:

Q: Define a relation on R by a ~ b iff aª – b² = bª – a². (a) Show that ~ is an equivalence relation.…

A: (a). To show that the given relation is an equivalence relation, we need to show that it is…

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

A:

Q: Let R be a transitive relation on set A. Define Q to be {(r Prove or disprove that (RUR®)n (QU Qº)…

A:

Q: Is it possible for a relation R on a set X to be both symmetric and antisymmetric? If so, give an…

A: Let R be a relation on set A .then SYMMETRIC relation: ANTISYMMETRIC relation: Recall: Identity…

Q: Let A= {1,2,3,4,5} and define a binary relation Ramong the subsets of A as follows: XRY XCY. a) Is R…

A: A relation 'R' is defined on set 'A' is said to be: Reflexive : ∀a∈A      aRa Symmetric: if aRb…

Q: Let R be a relation on the set A = {x € Z:1 < x < 100} defined as follows: x R y if and only if x-y…

A: Equivalence relation

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

A:

Q: Let ~ CX *X be an equivalence relation on a set X. For x EX define x := := {x |x E X} is a partition…

A: This is a question from Relation.

Q: Let A = {1, 2, 3, 4, ..., 22} and define a relation R on A as follows: For all x, y E A, x R y →…

A: Given A=1, 2, 3, 4, …, 22 and for all x,y∈A, xRy⇔4|x-y Hence, the relation R is given by…

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,

Q: 2. Define a relation R on Z as follows: a Rb if and only if 5 | (a²-b²). a) Prove that R is an…

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…

Q: Let A = {1,2,3, 4, 5, 6, 7, 8, 9, 10) and define a relation R on A by a R b if 6. and only if a and…

A: 6. By the definition, a R b iff a and b have the same remainder when divided by 4.     So, obviously…

Q: Prove that if R is an equivalence relation on a set A, then the inverse relation R−1 is an…

A:

Q: Let T be the relation defined by, For all a, b e Z, aTb iff a – b is odd. Is T an equivalence…

A:

Q: Let Z be the set of integers and R be the relation on Z x Z defined by: (a.x)R(b.y) if and only if a…

A:

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…

Q: Let A be a set and let ∼ be an equivalence relation on A. Let x, y ∈ A. Prove that [x] = [y] if and…

A:

Q: Define a relation on the set {a, b, c, d} that is 6. (a) irreflexive and transitive, but not…

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),…

A:

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

A:

Q: Define a relation Q on the set Rx Ras follows. For all ardered pairs (w, x) and (y, z) in Rx R, (w,…

A:

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…

Q: Côñšidéř thé rélátión S on the set {a, 0, C} given by S = { (a, a), (a, b), (b, a), (b, b), (c, c),…

A:

Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

A:

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

A:

Q: Use Gauss's Theorem to find the surface integral OF •dS across the positively-oriented S sphere of…

A:

Q: A relation R on a set A is backwards transitive if, and only if, for every r, y, z E A, if rRy and…

A: