If A is a subset of B and P(B)=0,then prove that  P(A)=0.

Q: Let A be a non-empty subset of R which is bounded below and let B be the subset of R defined by B =…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Show that if A is such that m (A)> 0, then there exists a bounded subset BCA such that m'(B) > 0.

A: By properties of Measurable set,         If then  .

Q: Let A be a nonempty subset of R. If a = sup A is finite, show that for each ɛ > 0, there is an a E A…

A: We have to prove the given statement

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

A:

Q: Show that if gcd (a, y) = 1 and a \ xy Then a\ x

A: Definition : Greatest common divisor ( gcd): Let  a and b be are two  integers, not both 0.  A…

Q: Let x E A. Prove that A is an infinite set if and only if A≈ (A - {x}).

A:

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

A:

Q: Let S and T be sets. Prove that SUT = T iff S is a subset of T.

A: To Prove S∪T=T iff S is a subset of T

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: Let S and T be subset of R. Prove the following: cl(SUT) = (cl S) U (cl T)

A:

Q: If S, T are nonempty bounded subsets of R with S ⊆ T, then inf T ≤ inf S ≤ sup S ≤ sup T.

A: Since S ⊆ T it follows taht inf T ≤ t, ∀ t ∈ T, then  inf T ≤s , ∀ s ∈S:Therefore inf T is  a lower…

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

A:

Q: 2. Let S is a subset of R. Prove that S is closed if and only if S contains all of its accumula-…

A:

Q: Show that if S x T is a closed subset of R2, then S and T are closed subsets of R.

A:

Q: Let X~U (0,0), then P(X > X²) is

A: Given that X~U0,θ fx=1θ-0=1θ, 0≤X≤θ

Q: Find c such that P(Z>c)= 0.3859

A:

Q: . Let A CQ satisfying the following conditions: (a) A + Ø and A # Q. (b) If p E A and q E Q with q <…

A:

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…

A:

Q: If S and T are nonempty bounded subsets of the real numbers, show that if ScT Then infT < inf S <…

A: Given ,  S is subset of T

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: 2) Let A be a finite set and a EA.Lf a EA is not a maximal element of A, then a has an îmmediate SUC…

A: Given A is a finite set. a∈A is not a maximal element of A. Now we have to prove 'a' has an…

Q: Prove that Z ≈ E*, where Z is the set of integers and E* is the set of positive even integers.

A: To prove:

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 if gcd (a, y) = 1 and a \ xy Then a \ x

A: if 1. gcd(a,b)=1then there exist some integers m and n such that am+bn=12. suppose a and b are…

Q: Suppose that |x| = n. Find a necessary and sufficient condition on r and s such that (x") C (x*).

A: Given,

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

A:

Q: Consider the subset W = {x € Q : xª > 1} of Q. 2.1 Express W using interval notation. 2.2 Is W…

A: W = x∈ℚ: x4&gt;1 This implies that W is a set of rational numbers 'x', such that x4&gt;1. if x4…

Q: Suppose T: R + P, has an action on a

A: Given, And

Q: Show that for every x E R, there exists an n EN such that <x<n.

A:

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

A:

Q: Suppose that S is a nonempty subset of R and k is an upper bound of S. The number k = sup S if and…

A: Given: S is a non-empty subset of R and k is an upper bound of S. To prove : Number k=sup S if and…

Q: For any measurable nonempty subset of R, L∞ (N) = I¹ (N).

A: Given: Measurable nonempty subset Ω of ℝ. To prove: For any measurable nonempty subset Ω of ℝ,…

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: Show that if gcd (a, y) = 1 and a \xy Then a\ x

A:

Q: Give an example of a closed subset S of R2 such that conv S is not closed.

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Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

A:

Q: Prove: f € R(a) if and only if for each ɛ > 0, there exists a partition of [a, b ] , say go, such…

A: f∈Rα if and only if for each ε&gt;0,there exists a partition of a, b,say P, such that UP, f, α-LP,…

Q: show that if Z ~ N(0, 1) and Y ~x²(n) with Z and Y to be independent, then T: n

A:

Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

A: By adding the cubes of positive integers we solve the given problem.

Q: If A and C are nonempty classes, prove that AC B and C C D if and only if AxC C BxD.

A:

Q: Let A and B be nonempty subsets of R such that A is bounded above and B is bounded below and let A –…

A: Let A and B are two non-empty sets of ℝ.Given A is bounded above and B are bounded below sets.Claim:…

Q: 12. If E is a bounded above subset of Z, then A. sup E exists and sup E E E B. sup E exists and sup…

A:

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: Let X be an infinite set. Define u : E = P(X) → Ř by %3D if A is finite µ(A) = 1 if A is infinite.…

A: Let X be an infinite set and ∑ =P(X) Let μ=P(X)=∑ → ℝ is given by μ(A)=0if A is finite1if A is…

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

A: Since P(E)=P(F), so the sample space of E and F are equal. Aso E is a subset of F so the elements in…

Q: Show that there must be two distinct subsets A, BC S such that

A: Every value in S is between 0 and 100 , So the sum of any subset will be at least 0 and  at most…

Q: 2. Write a preference E on R as a subset of R', where a bundle a E Rª is at least as good as a…

A: To write: A preference ⪰ on ℝ+2 as a subset of ℝ+4, where a bundle x∈ℝ+2 is at least as good as a…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.