3. Define the relation R on Z by mR, if and only if 2|(m – n). Show that R is an equivalence relation.

Q: (b) Is an equivalence relation on Z? If not, is this relation reflexive, symmetric, or transitive?…

A:

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: Suppose that R is a reflexive and transitive relation on a set A. Define a new relation E on A by…

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

A:

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…

A:

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: 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: Define the relation ~ on the set Z of integers by m~n if and only if m^2 = n. Show that ~ is not…

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: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

A:

Q: Let A = {0, 1, 3, 5, 7, 8} and R be a relation defined on A such that R= {(x , y)| 4 divdes (x –…

A: We are given the set A = {0, 1, 3, 5, 7, 8} and the relation R defined on A such that R = {(x, y)| 4…

Q: Let the relation non z defined by x ny if and only if 3/(x+2y) Show that n is an equivalence…

A:

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: F is the relation defined on Z as follows: Vr, n E Z, mFn =7|(m² – n²). (a). Prove that F is an…

A: Equivalence relation and classes

Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})2: y/x = 2k for some integer k}.…

A:

Q: S is an equivalence relation. 2. Define the relation R on Z as follows: for x, y E Z, rRy if and…

A:

Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…

A: According to the given information, Consider a relation R on R2 as:

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

A:

Q: Show that the relation R on Z defined by aRb if and only if 5a−3b is even, for a, b ∈ Z, is an…

A: Let R be a relation from a set A to itself. R is said to be an equivalence relation if 1) R is…

Q: 2. Consider the relation on the natural numbers N defined as follows: for all x, y e N, xRy if and…

A: According to question,We have to discuss about reflexive, symmetric, transitive relation.

Q: 4. For r, y E R, let z y if and only if (x- y) E Q. Show that defined as such is an equivale…

A: ~ is defined as : For all, x, y in R, x~y if and only if (x-y) belongs to Q Reflexivity : x~x for…

Q: 2. Let R be a relation defined on the set of integers Z where a R b if a +b is even. Determine…

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: 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: 9.15. A relation R is defined on Z by a Rbif |a – b| < 2. Which of the properties reflexive,…

A: Given : A relation is R is defined on Z is given by  a R b if a-b≤2  Which of the properties…

Q: Let S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an…

A:

Q: 1. A relation R is defined on Z* × Z* by (m,n)R(p, q) → m+q = n + p . (a). Prove that R is an…

A:

Q: A relation R on a set A is backwards transitive if, and only if, for every r, y, z E A, if rRy and…

A:

Q: Let C = {a, b, c, d}. Find a relation R on C that has exactly 3 ordered pair members and is both…

A:

Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})² : y/x = 2k for some integer k}.…

A:

Q: Consider the relation on Z×(Z\ {0}) defined by (m,n)R(m’,n’) provided that mn’ = m’n. Prove that…

A:

Q: 3. I Consider the relation R= {(r, y) | x+y is even} on the set Z of integers. Show %3D that R is an…

A:

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

A:

Q: A relation * is defined on the set R-{0} by x * y if and only if xy > 0. For each of the following…

A:

Q: Show that the relation R on Z defined by a R b if and only if 5a − 3b is even, for a, b ∈ Z, is an…

A: Given information: The relation R on Z defined by a relation b if and only if 5a − 3b is even, for…

Q: In this exercise, you want to show that a relation RC A? is transitive + R" C R for all n EN=…

A:

Q: Let S be the following relation on C10). S=[(x,y) E (C\[0]): y/x is real). Prove that S is an…

A:

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: 18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence…

A:

Q: Let R be the relation on Z defined by x Ry if and only if x+ 3y is even. Prove that R is an…

A:

Q: Define a relation on Z by setting xRy if xy is even. a. Give a counterexample to show that R is not…

A:

Q: Let R be the relation on N defined by a R b if either a | 2b or b | 2a. Is R an equivalence…

A: image is attached

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!