If P(E) = P(F) and E is a subset of F,then prove that E = F.

Q: Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects…

A:

Q: Prove that if a, b e Z – {0}, then gcd(a, b) = gcd(a, -b)

A: Solution:-   gcd a,b= Greatest common divisor of  a  and  b ∗Every integer  that divides  a  and  b…

Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

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Q: Let (X, F) be a o – algebra | , and be two probability measures on (X, F) so that both are ergodic…

A:

Q: Let E be a finite extension of R. Use the fact that C is algebraicallyclosed to prove that E = C or…

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Q: If F is a closed bounded set, then every infinite subset S of F has a limit point in F.

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Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: Let E be an extension of F and let a and b belong to E. Prove thatF(a, b) = F(a)(b) = F(b)(a).

A: Given Data E be the extension of F, and a & b belong E.   The E is an extension of F, and…

Q: 1. Let X be a nonempty set of sets. Prove that P(UX) = U{P(A)| A e X} if and only if UX E X.

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Q: What are the ASSUMPTIONS for t?

A: A t-test can be used when the hypothesis is concerned with the population mean and the population…

Q: Let (X, F) be a о — algebra , and be two probability measures on (X, F) so that both are ergodic for…

A:

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: If J= {x € Z|0 |O|. %3D Let F = {x € R | x5 is defined }. Then, F is a proper subset of R. If G =…

A: The given set is J=x∈Z: 0&lt;x&lt;4=1,2,3 So the number of elements of the set J is n=3. Hence the…

Q: If the Wronskian W of f and g is 3e^(4t), and f(t) = e^(2t), find g(t)

A: Wronskian W of f and g is given by:

Q: Prove that (a) P(En F) = P(E)- P(En F) (b) P(EnF) =1-P(E) – P(F) + P(En F)

A: Solution: 6. Here, it is needed to prove that

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: Let S be a non empty subset of R and let u e R be such that u + is an upper bound of S for each n e…

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Q: If X has full rank, so that P = X(X'X)-'X', prove that C(P) = C(X).

A: Given information: It is given that X is a full rank matrix.

Q: f:A→Z is not onto and f:A M is onto.

A:

Q: If S is a subset of T , then u* (T) su* (S).

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Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: Show that the mapping :S3 S3 given by p(x) = xis not a homomorphism.

A: Solution is below.

Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Prove that if H ≤ G, then aHa – 1 ≤ G, for all a in G.

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Q: Let P denote the set of functions f in C([0,1]) with f(x) > 0 for all x. Then P is a closed subset…

A: Given that, P denotes the set of functions f in C0,1

Q: || I(f) sup {Lp(f)| P is a partition of [a, b]} a

A:

Q: Prove that if x is not free in the formula A, then ` A ≡ (∃x)A

A: The formula is -------(1). To Prove: x is not free in the formula Proof: Let us assume that A(x) :…

Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: If f, g E R[a,b] with f < g on [a,b], prove that g.

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Q: Show that if x is in both W and W-, then x = 0.

A: Dot product of perpendicular vectors is 0.

Q: Let x, y, z E Z be such that both odd, and z is even. Prov

A: Introduction: Even and odd numbers are distinguished by the following characteristics: Even Number:…

Q: F(p), then prove that

A: simple use of integral transformation

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: If z, w E C, prove that z+w=z+w.

A: This follows from simple substitution. The detailed solution is presented in step 2. As requested i…

Q: Prove that if a, b E Z - {0}, then gcd(a, b) = gcd(a, -b)

A: soln:gcda,b=Greatest common divisor of a and bEvery integer that divides a and bsimultaneously will…

Q: Prove that if f is a continuous mapping of a set S (in E) into F and g is a continuous mapping of…

A: Given

Q: Let (X, F) be a о — algebra o - , and be two probability measures on (X, F) so that both are ergodic…

A: Let (X,ℱ) be a σ-algebra and μ≠v be a two probability measure on (X,ℱ) So that both are ergodic for…

Q: If S is a subset of T, then u*(T) su* (S)

A:

Q: Let z, w e C. Clearly prove that z+w = z + w.

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

Q: 2. Consider a measure space (X. A. u) and let s be a nonnegative measurable simple function on X.…

A: We have a measure space χ,Α,μ. Where A is a σ-Algebra and μ is a measure. We are given a function…

Q: Let f, g, h : N→ R²0. Prove or disprove that if f + g € O(h), then ƒ e O(h) and g E O(h).

A: Given f,g,h:ℕ→ℝ≥0 such that f+g∈Oh. We know that f∈Oh, if h is strictly positive for all large…

Q: Determine whether the function f: Z x Z -→ Z is onto if f (m, n) = .m + n

A: We will solve the problem using the definition of onto function.

Q: (a) Prove that if f e C (R²) then a (af af %3D

A: This is a problem of the calculus of several variables, higher dimensional calculus.

Q: Show that for all x, y E F, there exists e with e < Emachine such that fl(x * y) = (x * y)(1+ €) is…

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Q: Suppose that E is an alphabet, and that f :E* → E* has the property that f(ơ) = o for every o e E an…

A: Given information: Given that Σ is an alphabet. The given function is f.

Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

A: (a) Given that  f:X→Yg:Y→Z and g∘f is onto To prove : g is onto Let x∈X, y∈Y , z∈Z As gof is onto…

Q: If E and E, are measurable , then (E,UE) is not measurable

A: False.

Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.