Prove or disprove: 1. For any relations R and S, ran(S • R) = ran(S). 2. If R and S are equivalence relations on A, then S • R is an equivalence relation on А.

Q: 2. Consider the relation on A = {(1,0), (-1,0), (1,3), (2, 4), (-4,-8), (3,9), (3,6)}, defined by:…

A:

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 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: Consider the relation K on N defined by Knm iff n <m.Which statement below is true? K is an…

A: It is Irreflexive.

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: Show that “is unitarily equivalent to” is an equivalence relation on Mn×n(C).

A:

Q: Let Z be the set of integers. Consider R = {(r, y) e Z x Z: 7 divides x-y} CZ x Z. Show that R is an…

A:

Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

A:

Q: Disprove that for any relations R and S, ran(S • R) = ran(S).

A:

Q: PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² = u? + v?. Prove that…

A:

Q: Show that the relation R in the set A of all the books in a library of a college,given by R = {(x,…

A:

Q: 10) Which of the following statements is false? a) fR is reflexive, then RnRep b) ROR' p R is…

A:

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

A:

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

A:

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: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: 1. Let & be the relation of (a,b) la #b] on the set of integers. What is the reflexive closure of R?

A: As per our company guideline we are supposed to answer only one question with its 3 subparts. Here…

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-&gt;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

A:

Q: 3. Define the relation R on Z by mR, if and only if 2|(m – n). Show that R is an equivalence…

A:

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

A:

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…

A:

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: 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: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.

A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…

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: Given the following relation on {1,2,3,4}, is this relation equivalence? R = {(1,1), (1,3), (1,4),…

A: An equivalence relation on a set is a binary relation that is reflexive, symmetric and transitive.

Q: If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence…

A: Given, R1 and R2 are equivalence relations in a set A. So, both R1 and R2 are Reflexive, Symmetric…

Q: Consider the set Q = Z × (Z 10), and the relation fupon Q Defined as: (a, b) ↑ (c, d) + a•d = b•c a)…

A: we shall solve the part (a) only , if you want us to do other parts please resubmit the question…

Q: Show that the relation R on Z defined by aRb if and only if 5a−3b is even, for a, b ∈ Z, is an…

A: Let R be a relation from a set A to itself. R is said to be an equivalence relation if 1) 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 Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x R,…

A: The relation Q on the set R×R is defined as follows. For all ordered pairs w, x and y, z in R×R, w,…

Q: on K³defined by: (a, b, c) ~ (d, e, f) if and only if ak e K – {0} such that (a, b, c) = k(c, e, d)…

A:

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

A:

Q: Problem 17. (a) Prove that <r is a reflexive, transitive relation on P(E*). (b) Prove that =r is an…

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: 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:

Q: Determine, with justification, whether the following relations are equivalence relations. If so,…

A:

Q: Select the following relations on Z that are equivalence relations. ○{(a,b) ||a| = |b|} {(a,b) a²b²…

A:

Q: Is the relation aRb → ab < 0 an equivalence relation on the set Z? Justify your answer.

A: Given a relation, aRb⇔ab≤0. To check whether R is an equivalence relation on set ℤ. A relation is an…

Q: Let z be the set of integers and R be the equivalence relation on ZxZ defined by: (a.b)R(c.d) if and…

A: Given that R is a relation on Z×Z defined by (a,b) R(c,d) if and only if a+d=b+c.

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…

A:

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

A:

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: (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?).…

A:

Q: 1. Consider the following relations defined on the set {a, b, c}. For each relation, determine…

A: (a) Reflexive functions are the ones whose domain and range are the same  for instance a,a∈R1 So…

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

A:

Q: Let R be a relation on Z. Then R is an equivalence relation if it is defined by xRy if and only if O…

A: A relation R between the elements of a set is said to be equivalence relation if it is: Reflexive…