5) Suppose A={-3,-2,–1,0,1,2, 3} and R is defined on A by aRb ea? -b = 4k . Prove that R is an equivalence relation on A and identify the partition Pof A induced by R.

Q: c) For a set X = {1,2,3, 4, 5}, let P be the partition {{1,3, 4}, {2}, {5}}. In the equivalence…

A: Q1(c) asked and answered.

Q: With a binary relation R (defined below) on A and each tuple (a, b) indicating that a is related to…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: 3. Let the relation R on X = {1,3,5, 6, 8, 9, 11} be such that xRy means "x – y is a multiple of 3".…

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: 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: 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: The domain of the equivalence relation D is the set S: S = {7, 2, 13, 44, 56, 34, 99, 31, 4, 17}…

A: The domain of the equivalence relation D is the set S: S = {7, 2, 13, 44, 56, 34, 99, 31, 4, 17} For…

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: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

A:

Q: Consider the set A ={(1, 3), (2, 4), (-4,-8), (3, 9), (1, 5), (3, 6). Define the relation R on A as…

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: R is the congruence modulo 7 relation on Z: For all m, n E Z, mRn iff 7|(m – n). (i) Is the relation…

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: on P(X), where X = Y, Z E P(X), we say Y - Z if and only if Y has the same number of ele- Define a…

A: Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say  Y~Z if and only…

Q: Use a membership table to prove the following relation: * (An B) U (A n Bº) = A

A: We have to use membership table.

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: 1. Prove that “divides" is a transitive relation. That is, prove that, for all integers a, b, c, if…

A: I the given question we have to prove that , for all integers a,b,c, if a|b and b|c, then a|c.

Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

A:

Q: Let A = {-6, –5, –1, 0, 1, 5, 6}. R is defined on A as follows: For all (m, n) E A, m Rn 5| (m² –…

A: Given: A={-6, -5, -1, 0, 1, 5, 6} and relation R is defined on A as: ∀m,n∈AmRn⇔5|m2−n2 1. To show…

Q: 9. Let R be the relation defind on NxN by (m)R(m2) if and only if m1 – n1 = m2 – n2. Describe the…

A: The equivalence class of an element means a set of all those elements that are related to that…

Q: 2. Determine the partition of B = {1,2,3,4,5,6,7,8,9,10} induced by the equivalence relation B…

A:

Q: Let A = {1, 2, 3, 4, 5, 6} and R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (2, 3), (3, 2)}.…

A:

Q: Consider the relation R on the set A = {-6, –2,0,1, 4, 5, 6, 7, 15}, defined according to the…

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

A:

Q: Suppose A = {-4,-3,–2,–1,0,1,2,3,4} and R is defined on A by aRb a² - b² is divisible by 4. Prove…

A: A relation R on a set A is said to be an equivalence relation, if it satisfies the following three…

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: 5. Let A = {41, 49, 67, 71, 73, 76, 81, 88, 111} and R be an equivalence relation defined on A where…

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: 6. Consider the partition P = {{0}, {-1,1}, {-2,2}, {-3,3},{-4,4},...} of Z. Describe the…

A: Given the partition of Z is P=0,-1,1,-2,2,-3,3,-4,4,....

Q: 9. E is the binary relation defined on Z as follows: For all m, n E Z, m Enm-n is even. Is this an…

A: Equivalence relation

Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

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: Show that the relation R in the set A = {1, 2, 3, 4, 5} given byR = {(a, b) : |a – b| is even}, is…

A: Given that: A = {1, 2, 3, 4, 5} R = { (a,b) ; |a – b| is even}For any element a ∈A, we have |a -a| =…

Q: If {{1, 3, 5}, {2, 4}} is a partition of the set A = {1, 2, 3, 4, 5}, determine the corresponding…

A:

Q: Let A = {1,2,3,4,5} B= (6,7,8,9} Define an equivalence relation of p in A UB with atleast 12 members

A: A=1,2,3,4,5B=6,7,8,9A∪B=1,2,3,4,5,6,7,8,9

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: ) 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: (Chapter 11.4, #3) The relation relation. Sketch it, by drawing Z as a subset of the line, then…

A: The objective is to show that x≡ymod4 is an equivalence relation. To show the relation is an…

Q: 3) Let S be the equivalence relation on P({0, 1, 2, 3}) defined by XSY if and only if ged(X], 4) =…

A:

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: 4) Suppose R ={(a,b) €Z×Z:b-a is divisible by 3}, show that R is an equivalence relation on Z.…

A:

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 = {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: 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: (b) By the Equivalence Relation Theorem, there is an equivalence relation R on Z50 such that P is…

A: A relation from X to X is called a relation in X.  If R is in X and: xRx ∀x ∈ X ⇒ R is refelexive…

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: Let R be an equivalence relation on A. Find the maximum number of classes R can make if JA| = 80 and…

A:

Q: Suppose an equivalence relation R has the following equivalence classes that partition the set X.…

A: The ordered pairs in the equivalence relations produced by the given partitions {0, 2}, {1}, {3, 4}.…

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!