Answer the following questions about set theory and probability theory. a) Suppose the set X is a subset of the set Y. Is it the case that X is an element of Y? Explain your answer. b) Is it possible for there to be two sets X and Y such that X is a subset of Y, and (X ∩ Y) is empty? Explain your answer. c) Suppose X is a non-empty subset of Y. What is Pr(Y | X)? Explain your answer.

Q: (a) Prove that for events A and B, if P(B|A) > P(B) then P(A|B) > P(A).

A: As per our guidelines, we are allowed to answer first question only. Thanks   Given Events A and B…

Q: 9. If (X, M, ) is a measure space and E, FEM, then μ(E)+μ(F) = μ(EUF)+ μ(EnF).

A:

Q: For all sets A, B, and C, it holds that (An C) U (B \ C) = (B U C) \ (C \ A). True False

A: We have solve the set true and false

Q: Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a subset of B ? No.(Why?). Is B a subset of A?…

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Q: State whether the following claim is true and explain why. (a) Claim: For all sets A, B, and C: (A –…

A:

Q: Prove that if A and B are sets, then P(A) U P (B) C P(AUB).

A:

Q: Suppose that F and X are events from a common sample space with P(F) # 0 and P(X) #0. Prove that PX)…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Assume that AA, BB, and CC are subsets of a sample space SS with Pr(A)=0.65Pr(A)=0.65,…

A:

Q: S. is Setl

A: A small block slides without friction down an inclined plane starting from rest. Let sn be the…

Q: Consider the set Z = {A, B, C, D}.   Three items are selected from Z with replacement and…

A: Given data set Z={A,B,C,D}

Q: What can you say about the sets A and B if A ⊕ B = A?

A: The symmetric relation is described as, A ⊕ B =(A-B) U (B-A)

Q: There are only two states of the world, when a person is well with probability (1-p) and i with…

A: From the given information, A person is well with probability 1-p A person is ill with probability…

Q: Let A, B, and C be non-empty sets. Then Cx BxA=C×A×B if and only if O A=B O A=B=C O None of these O…

A: We can tackle this problem easily by assuming some examples  case (i) Let A=B=C = {1,2} we have,…

Q: Let A, B, and C be non-empty sets. Then A×B×C=A×C×B if and only if*

A: Let us consider four set A , B, C and D then AxC=BxD if all the elements of A are same as all the…

Q: " A is a subset of B. Therefore A is a subset of P(B)." יי

A: Since given statement is : " A is subset of B. Therefore A is subset of P(B)" Concept: First we have…

Q: Disprove the following statement by finding a counterexample. For all sets A, B, and 6, C, An (BUC)…

A:

Q: Show that if A, B, and C are sets, then|A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| +…

A:

Q: Let A, B, and C be non-empty sets. Then A×B×C=A×C×B if and only if

A: A, B, C are three non empty sets,  And, it is given that A × B × C = A × C × B

Q: A, IS thie set of all mes in the mple space that are not contained in A. B. The union of two events…

A: It is an important part of statistics. It is widely used.

Q: Show that for all sets A, B, and C, if A C BUC, then ANC + Ø or A C B.

A: In order to prove this we need to use the property that if X⊂(Y∪Z) ⇔X⊂Y or X⊂Z

Q: 3. Suppose that A is the set of sophomores at your school and B is the set of students in discrete…

A:

Q: If S is a subset of T , then u* (T) sµ* (S). ylgn O İhi O

A: False

Q: The Belmont report on ethics in research considers the following sources of harm Except: Physical…

A: The Belmont report considers the risk of potential harm.

Q: Write up a formal proof of the following: Claim: For any set A, A ⊆ A.

A: To prove rigorously that any set A is a subset of itself.

Q: Disprove: For all sets A, B, C, and D, (AUC) × (BUD) C (A × B) U (C × D).

A:

Q: (19) Prove or disprove ∀ sets A, B, and C: (A – B) − C = (A – C) – (B – C).

A: We will use the Venn diagram show the proof: Left side:

Q: Of the students at a certain university, 60% think their courses are challenging (C), 35% think…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Prove that, for any probability space (2, B, P) and events A, B E B, if Р () P(B) 1 P(AN B) " P(An…

A: If two events are independent then,

Q: Consider the following list of sentences about members of the set P of all people. 1. If anyone does…

A:

Q: If A, B, and C are sets such that A ∩ B ≠ Ø and B ∩ C ≠ Ø, then A ∩ C ≠ Ø

A:

Q: The lists below show five agricultural crops in Alabama, Arkansas, and Louisiana. Alabama Arkansas…

A: Set A:{s,p,c, hay, w}Set K:{s,r,t,hay,w}Set L:{s,sugarcane,r,c,t} Find A'∩K'

Q: 3. Suppose that A is the set of sophomores at your school and B is the set of students in discrete…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove the following statements about sets A and B. (AUB)° = A° Bº

A:

Q: 3. Using the table below, describe the following set in words and compute the cardinality: (Rn T) U…

A: According to the question: R= romance  T= Texas  H=horror So, (R∩T)∪H means "roman and Texas" or…

Q: Assume that A, B, and C are subsets of a sample space S with Pr(A) = 0.7, Pr(B) = 0.5, Pr(C) = 0.2.

A: note -Hi! Thank you for the question, As per the honor code, we are allowed to answer three…

Q: Prove or disprove that for all sets A, B, and C, we have a) A × (B ∪ C) = (A × B) ∪ (A × C). b) A ×…

A:

Q: Show that if A and B are sets, then P(A) u P(B) c P(A u B), where P(A) denotes the power set of A

A:

Q: The following table shows the results of a survey of authors by a fictitious publishing company.…

A: Consider the table: Describe the C∩N in words:C =The set of successful authorsN =The set of new…

Q: Let A, B, and C be non-empty sets. Then C×B×A=C×A×B if and only if * O None of these O A=C O B=c O…

A: A, B, and C are non empty sets

Q: find the subset of (E, B,T,*)

A:

Q: Alabama Arkansas Louisianasoybeans (s) soybeans (s) soybeans (s) peanuts (p) rice (r) sugarcane…

A: Given:U = {s, p, r, n, c, t, h, w}K = {s, r, t, h, w}L  = {s, n, r, c, t}To find: Kˈ∩ L

Q: Part 1: Proof that P(A N B) C P(A) N P(B). To prove part 1, select options from the list and put…

A:

Q: 1. If two balls are drawn from the box with replacement, compute (a, b, c, and d). 2. If two balls…

A: Given that a bag contains 5 balls, 3 are red and 2 are white.  Using this we have to solve parts 1,2…

Q: Let (N, E, P) be a finite probability space with sample space 52 = {a, b, c, d, e, and the event…

A:

Q: The probability of some events defined on the sample space Ω = {a, b, c, d} is defined as follows. P…

A: Given that Pc,d=3/8Pb,c=3/4   Pd=18 That is, Pc+Pd=38Pb+Pc=34Pd=18

Q: 1. If two balls are drawn from the box with replacement, compute (a, b, c, and d). 2. If two balls…

A: Given problem Given that   Red balls = 3 White balls = 2 Total balls = 5 1) If two balls are drawn…

Q: Consider the Venn diagram shown below.   Determine the following cardinalities. n(A)= ? n(C¯)= ?…

A:

Q: Let A, B, and C be non-empty sets. Then C×B×A=C×A×B if and only if * O None of these O A=C B=C A=B=C…

A: Given :- A, B and C are three non empty sets. To complete the statement. C*B*A = C*A*B if and only…

Q: 3 Prove or disprove each of the following: For all sets A, B and C, (a) (A – B) – C = (A – C) – B…

A: Solution is given below:

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.