a. Prove that the intersection of two equivalence relations on a nonempty set is an equivalence relation. p. Consider the equivalence relations R2 and R3 defined on Z by a R2 b if a = b (mod 2) and a R3 if a = b(mod 3). By (a), R1 = R2 N R3 is an equivalence relation on Z. Determine the distinct equivalence classes in R.

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

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

A:

Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

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 A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a)…

A: 11. Let A be a nonempty set and ~be an equivalence relation on A. (b)  We know that for each a, b∈A…

Q: R —A/R— is a set. You will need, of course, to find a superset of A/R, that is, a class X

A: To prove the given relation, show that as, reflexivity, symmetry, transitivity.

Q: Let m e Z+. Show that =m is an equivalence relation on Z. Prove that for any integer a,

A: Equivalence relation

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. Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible…

A: According to question given that Set Z in which the relation R is defined by aRbIf and only if a+3b…

Q: Let P be a set on which a binary relation < is defined such that, for all r, y, z € P, (a) * <x is…

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

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: Let A = {82,95, 146, 153, 158, 166, 180, 187, 207} and R be an equivalence relation defined on A…

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: 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: Let A be a non-empty set and R be an equivalence relation on A where a is an element of A. Define…

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: . Define r on the power set of {1,2, 3} by ArB → |A| = |B|. Prove that r is an quivalence relation.…

A:

Q: Define the relation on the set Z of integers by if and only if a ~b if and only if a = b + 5k for…

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: b

A:

Q: Suppose that G and H are equivalence relations in A, and that G C H. Prove that Gr Gy if and only if…

A: It is given that G and H are equivalence relations in A and G⊆H. The objective is to prove that Gx…

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

A:

Q: Let S' = {x € R² : ||x|| = 1} C X. and endow S' with the same equivalence (b) relation - as above.…

A:

Q: 9.29. Let R be an equivalence relation defined on a set A containing the elements a, b, c and d.…

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

A:

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: 11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following:…

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: 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. 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 S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: Which of the following statements are true about an equivalence relation R on a set A? (Select all…

A: .

Q: let R be an equivalence relation and let [x] be the equivalence class of x, prove x belong to [x]

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

A:

Q: Let A = {55, 63, 70, 83, 86, 106, 113, 116, 151} and R be an equivalence relation defined on A where…

A: Given : A=55, 63, 70, 83, 86, 106, 113, 116, 151 and R is an equivalence relation defined on A where…

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

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Q: #1. Let R be the relation defined on the set of all integers Z as follows: for all integers m and n,…

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

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Q: 1. Let X be a set, R, S C X × X binary relations on X. Prove or disprove the following. (a) If R and…

A: We have to solve given problem:

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:

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.