(3) In each item below, verify that - is an equivalence relation on the given set. (c) Let - be a relation defined on Z by a Rb if and only if a' = b3 (mod 4). Determine the distinct equivalence classes determined by this equivalence relation. (Hint: There are only three distinct such congruence classes.)

Q: (a) classes of X induced by Prove that - is an equivalence relation on X and describe the set X/ ~…

A: This is a problem of relation.

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: , In each item below, verify that - is an equivalence relation on the given set. (b) Let - be a…

A:

Q: 1. Let H = {1, 2, 3, 4, 5} and the rlation RC H², with (a, b) ER + a = b( mod 3). • Give the set R.…

A: R ={(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}

Q: Let R be a relation on the set of all integers such that aRb if and only if 3a – 5b is ever 1) Is R…

A:

Q: 2) Let R be binary relation on N defined by r Ry if and only if r <y< 2r. Is R reflexive? Is R…

A:

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: Define a binary relation R on the set A = {binary strings of length 6} as follows: For all strings…

A:

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…

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A: Consider the provided question, Since, A relation R that is reflexive, anti-symmetric and transitive…

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

A:

Q: Suppose R is both an equivalence and partial order relation. Show that every equivalence class of R…

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: 4. Define a relation on the set N of natural numbers as follows: For natural numbers n and m, n ~ m…

A:

Q: b

A:

Q: 10. (A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

A:

Q: Prove that "is similar to" is an equivalence relation on Mnxn(F ).

A: We have to prove that “is similar to” is an equivalence relation on Mnxn(F ) i) Every matrix A is…

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: (1) Let R be the relation defined on Z by aRb if 2a + b = 0 (mod 3). (a) Prove that R is an…

A: (a) The relation between a and b represents that 2a+b is a multiple of 3.

Q: 9.45. A relation Ris defined on Z by a Rb if 3a + 56 = 0 (mod 8). Prove that Ris an equivalence…

A: 3a + 5b ≡  0 (mod 8)

Q: 7. Recall that, for n e Z, n > 0, there is a relation on Z called congruence mod n. The b (i.e.…

A:

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: Let N be a nonempty set and o a permutation of N. Define a relation a~b if and only if b = o"(a) for…

A: Given, Ω is a nonempty set and σ a permutation of Ω. A relation on the set is defined as a~b if and…

Q: 9.51. Let R be the relation defined on Z by a R b if 2a + 3b = 0 (mod 5). Prove that R is an…

A:

Q: (2) Let R be binary relation on N defined by rRy if and only if r <y< 2r. Is R reflexive? Is R…

A: We have given that R be a binary relation on ℕdefined by xRy if and only if x≤y≤2x. Now we have to…

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…

A:

Q: Let ~ be defined on the set X = {0,1,2,3,4,5,6,7,8,9} by x ~ y + x2 = y? mod 5. Prove that is an…

A: Equivalence relation are those which are reflexive symmetric and transitive

Q: 2. Consider the relation on the natural numbers N defined as follows: for all x, y e N, xRy if and…

A: According to question,We have to discuss about reflexive, symmetric, transitive relation.

Q: 2. Let R be a relation defined on the set of integers Z where a Rb if and only if a? = b° (mod 5).…

A:

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: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…

A: Consider the given information.

Q: For every nonempty set A, there exists a relation R which is both an equivalence relation and a…

A: A relation R is said to be equivalence relation if R is Reflexive ,Symmetric and Transitive . R is…

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: ~ on R? by (x, y) ~ (u, v) if and only if ? + y? = u² + v2. In class, we saw that this defines a…

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: 3. a) Prove that the relation~ defined by A~ B if and only if A and B are conjugate is an…

A: As per our guidelines we are supposed to answer only three subpart. Here first 3 sub

Q: 1. Given the binary relation R on R defined by „R, if and only if x > y – 1, for each of the…

A:

Q: Define - on Z as follows. Suppose that a ~ b if a² = b² (mod 6). Prove that - is an equivalence…

A: We need to prove 1) Reflexive 2) symmetric 3) transitive.

Q: . Determine whether or not each of the binary relations R defined on the given sets A are reflexive,…

A:

Q: 14. For a,b E Z, define a ~ b if and only if 4a + b is a multiple of 5. Show that defines an…

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: (c) Let ~ be a relation defined on Z by a R b if and only if a’ = b³ (mod 4) . Determine the…

A:

Q: Question 7 Define a relation R on the set of integers Z by: aRb if and only if ab >0 (a) Show that R…

A:

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: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…

A: A relation to be an equivalence relation must satisfy the following three properties:  1. Reflexive…

Q: Define a relation R on the integers Z saying that (m, n) is in R if m2 is equivalent to n2 (mod 7).…

A: First to prove the relation, R, on set of integers Z is equivalence relation. R is defined as:

Q: 3. Prove that a = b(mod7)is an equivalence relation by taking suitable examples.

A:

Q: Let R be the relation on the set of integers defined as aRb + 5a + 8b = 0 (mod 13). (a) Show that R…

A: