6) Let A = {1,2,3,4}, show that by counterexamples that the following relations are not equivalence relations on A a) xRy xy is even. b) xRy > x- s1 c) xRy x divides y

Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

A:

Q: 4. Consider the following binary relation p on the plane R² (x, y)p(u, v) if y? – a² = v² – u².…

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Q: Let A = {1, 2, 3}. Then show that the number of relations containing (1, 2)and (2, 3) which are…

A: The smallest reflexive relation on set A containing (1, 2) and (2, 3) is

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

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Q: PROBLEM1 Instruction: Prove that the relation is  an equivalence relation.  For x, y ∈ R say x is…

A:

Q: 4. a) Define the following terms: i) POSETS ii) Equivalence relation

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 4. For (1, yı) and (r2,y2) in R² we set (11. yı) ~ (12. Y2) →{+ =3+ yi. (a) Show that ~ defines an…

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Q: PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² = u? + v?. Prove that…

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Q: Use the Ruler axiom to prove that there is a one-to-one correspondence between any two lines.

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Q: We defined the relation between sets by A ~ B means that there there is a 1-1 correspondence ƒ : A →…

A: We define the relation ~ between sets by A~B means that there is a one-one correspondence f:A→B Show…

Q: Let A = {a, b, c, d, e, f,g, h}, and let G and H be the following equivalence relations in A: %3D G…

A: Given that A = a , b , c , d , e , f , g , h . Also,  G and H are the following equivalence…

Q: 9.19. A relation Ris defined on Z by x Ry if x · y > 0. Prove disprove the following: (a) R is…

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Q: Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :P1 and P2 have same…

A: Given, relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number…

Q: Show that the relation R defined in the set A of all triangles as R = {(T1, T2) :T1is similar to…

A: Given,relation R defined in the set A of all triangles as R = {(T1, T2) :T1 is similar to T2}.

Q: Show that if AB BA, then eA+BeAeB, but if they do commute, then the relations hold

A: The problem concerns exponentials of matrices and commutativity

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: The relation by a = b (modn) is Partial Order Relation Ture O False O

A:

Q: Show that the number of equivalence relation in the set {1, 2, 3} containing(1, 2) and (2, 1) is…

A: To show:The number of equivalence relation in the set {1, 2, 3} containing (1, 2) and (2, 1) is two.

Q: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…

A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric)  So, R∘S=S∘R

Q: Given the following relations on the set {1, 2, 3} and R = {(1, 1), (1, 2), (2, 1), (1,3)}. Which of…

A: Given problem:- Given the following relations on the set {1, 2, 3} and R = {(1, 1), (1, 2), (2, 1),…

Q: Let Let A={2,4} and B = {1,3,5} and define relations U, V, and W from A to B as follows. For all (x,…

A: Given that -  Let A={2,4} and B = {1,3,5}

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…

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Q: Prove and say the name of the following equivalence relation x ~ y ⟺ x - y = 4k, k ∈ Z

A: x ~ y iff x - y = 4k , k in Z Implies that  x is congurent to y mod 4.

Q: 1 If MR =: 1 lo 1 1., then the relation ol 1 Ris Reflexive Transitive Anti-symmetric Symmetric

A: To find: The type of relation R. Concept used: Reflexive: If all the diagonal element of MR is 1,…

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: 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: Show that given relation is transitive For X, y € R say x is congruent to y modulo Z if x – y is an…

A:

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: Let the relation R defined on the set of positive integers as (x, y) ER if 3 divides X - y Determine…

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Q: Show that the “Relation is Congruent to” on the set of all triangles in a plane is an equivalence…

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Q: 6. Find all equivalence relations on {1,2,3}. 7. Show that for n, m ɛ Z+, gcd(2" – 1,2m – 1) =…

A: Note : We are entitled to solve only one question at a time and up to 3 subparts only. To find -…

Q: Let A = {4, 5, 6} and B = {5, 6, 7} and define relations R, S, and T from A to B as follows: For all…

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

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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. Prove that the next relation is equivalence relation on the set of all people: {(x, y)|x and y…

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

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Q: Which of the following are equivalence relations? If they are not, which of the axioms fails? a) For…

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

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Q: Use set builder notations to prove that:  (a) X ⊕ Y = (X ∪ Y ) − (X ∩ Y ),           (b)(Y −X) ∪ (Z…

A: We have to prove the following using the set builder notation: (a) X⊕Y=(X∪Y)−(X∩Y) (b)…

Q: Obtain the 16 relations of the set {0,1}, indicate which of these are reflexive, symmetric and / or…

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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 S = P({1,2,3,4,5}.. Define equivalence class of the set {1,2,3}. equivalence relation ~ by X~Y…

A: Consider the given information:

Q: Prove that the following relation is equivalence relations and prepare it in latex: On R, with x∼y…

A: Revision : A relation ~ defined on a set  A is an equivalence relation if and only if it satisfies…

Q: This question refers to unions and intersections of relations. Since relations are subsets of…

A: The given sets are: A={-2,2,3,5} and B={2,3}

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…

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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: (b) Let R and S be two equivalence relations on X. Are RnS, RUS, R \ S, also equivalence relations…

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Q: 9.16. Let A = {a, b, c, d}. How many relations defined on A are reflexive, symmetric and transitive…

A: given A=a,b,c,d claim- how many realations defined on A are reflexive , symmetric and transitive and…

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!