37. Assume that IT is a partition ofa set A. Define the relation Rn as follows: Show that Rn is an equivalence relation on A. (This is a formalized version of the discussion at the beginning of this section.)

Q: Suppose R is a relation defined on the set A = {1,2, 3, 4}. If R is an equivalence relation, what is…

A: Given below the detailed solution

Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

A:

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

A: S=0,1,2,3×0,1,2=0,0;0,1;0,21.0;1,1;1,22.0;2,1;2,23.0;3,1;3,2

Q: 4. Let f : X → Y define a relation on the set X by „Ry if and only if (x) = fy). Is f an equivalence…

A: f is an equivalence relation if f satisfies three properties. 1. if f is a reflexive relation  2. if…

Q: Let R be the relation on the set of integers, s.t., aRb if and only if a=b or a= -b. Is R an…

A:

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

A:

Q: Given a set define a relation induced by a partition of set A and that partition must consists of at…

A:

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: Determine whether the following relation is an equivalence relation? R= {(a, a), (a, c), (c, a), (b,…

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: 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…

A:

Q: 9.24. Let R be an equivalence relation on A = {a, b, c, d, e, f,g} such that a Rc, c Rd, d Rg and b…

A:

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

A:

Q: Give a specific reason why the following set R does not define an equivalence relation on the set…

A: We will find out the required value.

Q: efine the relation ~ on the set N of positive integers by a~b if and only if  a = b(10^k) for some…

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: Consider the equivalence relation ~ on R given by a ~ b if and only if [a] = [b]. You do not need to…

A: Given that the equivalence relation ~ on ℝ defined as a~b if and only if a=b. Define a function…

Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

A:

Q: (b) Observe that all the relations in (a) are equivalence relations. Prove that this is true in…

A: Given: A is a nonempty set, and P is a partition of A. To show: The relation R is equivalence…

Q: Subject: Set Theory

A:

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

A:

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: be an equivalence relation on the set A = {1, 2,...,8}, and denote the equivalence class of x E A by…

A:

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: 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…

A:

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: Let A = {a, b, c, d, e} and define an equivalence relation R C A × A on the set A as follows: R =…

A: We will answer the first question as you didn't specify any. Please resubmit the other question…

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

A:

Q: Determine whether or not the following relations are equivalence relations on the given set. If the…

A: Let m, n, p∈ℚ Relation is m~n in ℚ if mn≥0 (i) Reflexive: Let m∈ℚ then mm=m2≥0 Therefore m~m   ∀m∈ℚ…

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: 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: 1. Show that R is an equivalence relation on P(A). 2. Set A = {1, 2, 3, 4} and B = {1, 24}.…

A:

Q: Consider the subset relation C on the power set of {a,b,c}, that is P{a,b,c}. (a) Construct the…

A: Let S be a partially ordered set. Use the following definitions, to obtain the solution. Let a,b∈S…

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

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: Prove that the relation is an equivalence relation. 1. For x,y E R say x is congruent to y modulo Z…

A:

Q: Draw the Hasse diagram for the relation C on the set P({1,2,3}).

A: According to the given information, it is required to draw the hasse diagram for the given relation.

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: Let A # 0 be a set. Prove that A? is an equivalence relation on A.

A:

Q: Determine whether or not the following relations are equivalence relations on the given set. If the…

A:

Q: 2. Let H ≤ G and define = on G by a = b iff a¯¹b € H. Show that =µ is an equivalence relation.

A:

Q: 10. Let be the relation defind on Z by „R, if and only if a|b. Orve an explicit description of the…

A:

Q: R How to prove that " If is the equivalence B relation on the set and is the function R = IB then

A: given in question R is a equivalence relation on set B that means R is reflexive ,transitive  and…

Q: Let S be a nonempty set. Consider the relation ~ on P(S) defined byA ~ B if A U BC = S. Is the…

A: Let S be a nonempty set. Consider the relation ~ on P(S)  defined by A~B if A∪Bc=S To show: "~" is…

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…

A:

Q: Prove that the given relation is an equivalence relation, and describe the distinct equivalence…

A: Let A is the set of all students at your college. For every x,y∈A,  xSy⇔x is the same age as y. That…

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.