Consider a relation R on Z- {0} defined by the rule that (x, y) E R if and only if xy > 0. a) Prove that R is an equivalence relation. b) Determine all elements in the equivalence class containing 1. How many distinct equiva- lence classes are there?

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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.

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Q: 8. Let A = Z* U {0}, define relation R on A × Z* by (k, l)R(m,n) → kn = lm. (a). Prove that Ris an…

A: In part (a) we use the definition of equivalence relation i.e., reflexive, symmetric and transitive.…

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

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Q: Let R be the relation consisting of all pairs (x, y) such that x and y are strings of uppercase and…

A: Let R be a relation on a set A R is an equivalence relation if it is reflexive, symmetric,…

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

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

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Q: - b if and only if f (a) = f(b). Show that ~ is an Let f: A → B be a surjection. Define a ~…

A: Prove ~ is an equivalence relation. 1) Reflexive Since f is a function, f(a) = f(a) and so a ~ a for…

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

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Q: Define a relation R on Z by declaring that xRy if and only if x^2 =y^2 (mod4). Prove that R is…

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Q: Consider the equivalence relation on R defined by r ~ y if x – y € Z. Describe the quotient space…

A: In topology and related areas of mathematics, the quotient space of a topological space under 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)…

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Q: Define a relation " on Z by a ~ b if 3a + 4b = 7n for some integer n. Prove that - defines an…

A: A relation ~ on ℤ by a~b if 3a+4b=7n for some integer n.

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…

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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: Let A = {0,1,2} and r = {(0,0), (1,1), (2,2 y that r is an equivalence relation on A.

A: The relation is reflexive, since a,a∈R, where a∈A Also, since a=b where (a,b)∈R therefore, the…

Q: 1. Prove or Disprove: For any set A, there exists a relation R on A such that R is both symmetric…

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Q: In the Cartesian plane = {(x, y): x,y E R} consisting of points with rectangular coordinates (x, y),…

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Q: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…

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Q: Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3). (a) Prove that R…

A: (a) for any x in Z,       obviously xRx, as x2 = x2 mod 3       Hence, R is reflexive.        If…

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: ) Let A = {1,2,3, 4} × {1,2, 3, 4}, and define a relation R on A by (1, Yı)R(x2, Y2) if r1 + y1 = x2…

A: Equivalence relation means it should satisfy reflexive, symmetric, Transitive conditions.…

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…

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Q: Let A = {2,4,6,8,10}. The distinct equivalence classes resulting from an equivalence relation R on A…

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Q: Let R be an equivalence relation on A - {a,b,c,d) such that a Rc and bRd How many distinct…

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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…

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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)…

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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: . 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: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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

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Q: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

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Q: Consider the binary relation R = {(x,y),(x,z),(z,x),(z,y)} on the set (x,y,z}, which one of the…

A: We have to find which one of the options is true for relation R.

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: 1. Let - be a relation over Zx (Z\ {0}) defined by (a,b)-(c,d) iff ad = bc. Prove that this is an…

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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…

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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: 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: 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…

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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: 40

A: By using the definition of equivalence relation solution is given as follows :

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

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!