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

A:

Q: 2. Let A be any nonempty set. (a) Give the definition for: "A is not countable." (b) Is it true or…

A:

Q: Claim: More than 3.1% of homes have only a landline telephone and no wireless phone. Sample data: A…

A: Z-test for one population proportion is a test that helps us compare a population proportion with a…

Q: -8. a. Show that if A and B are independent, then P(A|B) = P(A). b. Say we have sets A, B, and C (do…

A:

Q: Consider two sets, A and B. (a) Prove that AnB = if and only if A - B = A. (b) Prove that A = (ANB)…

A:

Q: Prove or disprove. Let a, b, c ∈ Z. If a|c then ab|c

A:

Q: Let A (k,s), t.en the power set of A is:

A: Set of all subsets of A is called power set of A.

Q: Given A = {3, 6, 9, 10, 12} and B = {1, 6, 9, 12}, find each of the following set cardinalities:…

A:

Q: Define the following sets:A={x, y, z, t}B={s, t, u, v, w, x}C={t, v, u, y}D={s, t, u, v}Determine…

A: Given: A=x, y, z, t, B=s, t, u, v, w, x, C=t, v, u, y, and D=s, t, u, v. To determine: Following…

Q: Prove the statement: Let A be a subset of a nonempty finite set U. If P(A)=a, then P(Ā)=1-a.

A: We have to prove given statement...

Q: Let A and B be two measurable sets prove that if ASB then BIA is measurable and m(B\A)=m(B)-m(A)

A: The given problem is related with measure theory. Given that A and B are two measurable sets.  We…

Q: Negate the following (in English): a) If it is snowing, then it is cold. b) If x E (0, 1), then x2 <…

A: When we are to negate the statement of the form "If A then B" then the statement should be replaced…

Q: Draw a Venn diagram that represents (S ∪ T) − (S ∩ T).

A:

Q: Discrete Math problem: Prove that if X∩Y = X∩Z and X∪Y = X∪Z, then Y=Z for all sets X,Y, and Z.

A:

Q: Probability spaces: Consider the state space N = {w1,w2, w3, wa}, with associated o algebra F = 2",…

A: Given information: Given data represents two probability measures P and Q.

Q: Let X = {a,b,c,d}. List the elements of the set {A :A € P(X) and |A| = 2}.

A:

Q: Let F be an event space. (a) Show that A\ B € F whenever A, B E F.

A:

Q: Two sets A and B are nonempty disjoint subsets of a set S. If x e S, then which of the foll- (a)…

A:

Q: Suppose that the universal set U = { 1,2,3,... 20 } and that sets S and T are defined as follows: 1.…

A:

Q: 2. Let A = {0,1, 2, 4, 5, 7} and B = {2,3, 4, 5, 8} be subsets of the set %3D %3D U = {n € Z | 0 < n…

A: Given: A=0,1,2,4,5,7  &amp; B=2,3,4,5,8 is subset of U=n∈ℤ  0≤n≤9 i.e.U=0,1,2,3,4,5,6,7,8,9 To Find…

Q: Answer the remaining questions about set theory and probability theory.   d) Give an informal…

A: Given  X⊂ Y  Y⊂ X

Q: (d) How many subsets V of A are there so that 1 € V or VOB= Ø? Explain.

A: This question related to set theory . Solution is given below.

Q: Given the sets U = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } , A = { 0 , 2 , 4 , 5 , 9 } , B = { 1 ,…

A: Given that the universal set is U=0,1,2,3,4,5,6,7,8,9. Also, the sets A and B are given by…

Q: A Partition of a set has 3 subsets: A, B, and C. If Pr[A]=.25 and Pr[B]=.45, find Pr[C].

A: The provided information is,P(A) = 0.25P(B) = 0.45According to the general law of probability, the…

Q: Suppose a family contains two children of different ages, and we are interested in the gender of…

A:

Q: Let the universal set U = {Meansville, Molena, Stockbridge, McDonough, Hampton and Locust Grove,…

A: Given that,  The universal set is: U=Meansville, Molena, Stockbridge, McDonough, Hampton and Locust…

Q: 3. Given a set X and subsets A and B of X find A x B, A U B, AB, A-B, B-A, A, Bc. a.)…

A:

Q: 2. Which of the following shows the collection of all outcomes that are elements of one or the other…

A: 2. A. A ∪ B3. A. EventExplanation:2. The collection of all outcomes that are elements of one or the…

Q: Consider the following statements about subset relations between sets: i. N ⊆ Z ii. R ⊆ N iii. Z⊆R…

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

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”.