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

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

A:

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

A:

Q: Let R be a relation on the set N of positive integers defined by aRb if and only if the product axb…

A: This question is about equivalence  class .

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: 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: 1. (p.209, #16) Define a relation R on Z by declaring that xRy if and only if x? = y? (mod 4). Prove…

A:

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

A:

Q: SULU Let R be a relation over the integers (Z). Prove that R= {(a,b) : a mod 4 = b mod 4} is an…

A:

Q: Define R on Z+ by aRb if = 5 for some integer k. (a) Prove R is an equivalence relation. (b) What is…

A: The solution is as follows:

Q: 3. Let R be the relation defined on Z by aRb if and only if 5 | (4a + b). Prove that R is an…

A: We have given relation on ℤ and defined by  aRb if and only if 5|(4a+b). We can  see here for any…

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: Let E be an cquivalence relation on Z defined by mEn iF and only iF (mtn) is even, where m,n € Z. IS…

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

A:

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: Consider the equivalence relation {(r, y) | a = y (mod 6)}| on the set {n e Z| – 9 <n< 15}| Use the…

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: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

A:

Q: b

A:

Q: Let D be the binary relation defined below. For all u, w e 2, u D w → w=u +7 or w= u - 7. Show…

A: Solution of given problem is as below: A binary relation is equivalance relation if it satisfies the…

Q: Let R = { ( a,b ) | a,b ∈ Q and a − b ∈ Z } . Prove that R is an equivalence relation on Q

A:

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: 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: 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: 6. Consider the relation R on Z defined by xRy iff x – y= 4n for some n Ɛ Z. (a) Show that R is an…

A:

Q: Given a natural number c ∈ N. On natural numbers, the relation Rc is defined as follows: ∀ a, b ∈ N…

A: (a) let (a,a) be in Rc .      Then there exist integers u and v such that       au + av = c…

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: on K³defined by: (a, b, c) {0} such that (a, b, c) = k(c, e, d) is an equivalence relation on K3,…

A:

Q: 10. (1) A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

A:

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: 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: Let A Z and let the relation R be defined on A by zRy if and only if 7 divides r-y. a) Show that R…

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

A:

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

A:

Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in  R…

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

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: Define a relation R on Z by declaring that xR y if and only if x^2 ≡ y^2 (mod 4). Prove that R is…

A:

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

A:

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

A:

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: 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:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.