Let A be a set. Prove that S = 0. SEP(A)

Q: j) A U A = A k) A U $ = A

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Q: Let A be a set. Suppose that f: w→ A is onto A. Prove that A is countable.

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Q: If A and B are bounded subsets of R, then prove that AUB and AB are bounded.

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Q: LAUBUC = ANĒNT. %3D

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

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Q: Find an example of a closed convex set S in R2 such that its profile P is nonempty but convP ≠ S.

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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: Let E CR be a closed set. Prove that aE CE.

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Q: Let A be a compact set in R. Show that the complement A is open and contains an interval of the form…

A: Solution: Before going into the problem we must know the Theoem named Heine-Borel Theorem.…

Q: For any set S c R, let 5 denote the intersection of all the closed sets containing S. (a) Prove that…

A: Consider the given information.

Q: Prove that any closed interval [s, t] can be written as the intersection of a collectable collection…

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Q: Show that the set (1) isopen u k = 1 and find its measure.

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Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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Q: (a) Investigate whether d: RxRR delined as d(z, u) = (- u) is a metrie on the set ol all real…

A: We investigate given distance function is metric on R or not

Q: m) Show that if T;,T; and Ty are topologies on a set X.then r, nnt, is a topology on X.

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

Q: Let (x, eA) be a measurable space, £: X (-∞, w) and 9:* (-∞0, ∞o) are measurable. Prove that the…

A: Given that X,A is a measurable space. The objective is to prove that the sets  {x∈X:f(x)<g(x)},…

Q: Call a set A CR clopen if it is both open and closed. Prove that the only clopen sets are Ø and R.

A: A⊆R clopen if it is both open and closed.  We have to prove that the only clopen sets are ϕ and R

Q: the collection of all close subsets of R is called the topology ofR ylgn İhi

A: the collection of all close subsets of R is called the topology of R

Q: Let Ø denote the empty set .Prove each of the following. (a) (A-B)NCACUB)=0 For all sets A and B.

A: Union and Intersection of Sets: The union of two sets A and B is the set of all elements which…

Q: Let T and M be a non-empty families of sets with T  ⊆ M. Show that ∩T  ⊆ ∪ M.

A: Given: T and M are families of sets with T⊆M. To show: ∩T ⊆∪M

Q: Let E be a nonempty subset of an ordered set; suppose a is a lower bound of E and B is an upper…

A: Let X be a non empty set. Let a∈X. Then a is said to be a lower bound of the set X if a≤x for all…

Q: Show that E is a measurable set if and only if for every E > 0 there is a closed set F and an open…

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Q: Let X be a set and {Ya|a E A} an indexed family of sets. Prove that X U (Naea Ya) =Naea (X U Ya).

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Q: A. Show that T is a topology in X and Show that B,(x) is open set with respect to t for all r > 0

A: This is a problem of Topology.

Q: Let A and B be sets. Let X = A n B, Y = A \ X and Z = B\ X. Prove that X n Y = Ø, X n Z = Ø, Y n Z =…

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Q: Let X be an infinite set with the finite closed topology T={S subset of X; X-S is finite}. Then

A: a) If not, then there are non-empty disjoint open sets  such that  . Since they are disjoint, so  .…

Q: Let X = R (or any uncountable set). Define µ* : P(X) → R by { if A is countable µ*(A) = 1 if A is…

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Q: 2. Let X be any uncountable set, Prove that: 7. = (GC X: G is countable} U {0} is a topology on X.

A: Since you have posted multiple question... according to company rule we are supposed to answer first…

Q: Let A be a subset of some universe U. Prove that AN A° = Ø.

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Q: Show that Let ECR" is measurable set. If µ(E) >0, then E have a non-measurable subset

A: Here, we introduce the notation Gr :=G∩[0, r) + 1-r ∪ G∩[r,1) - r for G⊂[0,1) and r∈R. Note that…

Q: Let T and T 'be two topologies of a set X. Is the family T U T´formed by the openings common to both…

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Q: Let (E), be a countable disjoint collection of measurable sets. Prove that for any set A, 00 m* (AnU…

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Q: Let $X$ be a set and assume that p E X. Let T, be the collection (Ø, X, and all subsets of X that…

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

Q: Show that for any nonempty set A, there exists unique set B such that for any x. xEB ex belongs to…

A: From your question, it is clear that "A" is a set of sets, i.e., members of the set "A" are sets.

Q: consider R with the usual topology then a subset A={1,0.5,..} is: empty or closed or open or none…

A: Given R with the usual topology and given a subset  A=1, 12, 13,.....

Q: Find an example of on a set X (G has or H of G which has le

A: Given: that Group G is active transitively on set X . G has only one orbit and subgroup H of G has…

Q: Let {An : n E N} be the indexed collection of sets defined by 1 2, 2+ 2n An Prove that: nMeN An = Ø

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Q: Let {Aα:α∈I}{Aα:α∈I} be a nonempty indexed collection of sets.  Prove or disprove…

A: Let Aα:α∈I be a non empty indexed collection of sets.  If A is a family of sets, the intersection…

Q: 3. Let a, b e R such that a < b. Prove that the open interval (a, b) is an open set in R.

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Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

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Q: Let X be an infinite set and T be a topology on X. If every infinite subset of X is closed, then T…

A: Given that Let X be an infinite set and T be a topology on X.If every infinite subset of X is…

Q: Let $X$ be a set and assume that p E X. Let Tp be the collection 0, X, and all subsets of X that…

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Q: )Š is the smallest closed set in M such that S CS.

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Q: Suppose A is a nonempty subset of R. (a) Is it true that Là=L? Prove or disprove. (b) Let iso(A)…

A: Given that A is a non-empty subset of ℝ. (a) To prove that                      LA¯ = LA

Q: the collection of all close subsets of R is called the topology of R ylgn İhi

A: False

Q: Let X be an n-element set. Prove that there are 3n ordered pairs (A, B) of subsets of X such that A…

A: Solution: Let us assume that the subset A has j elements.After fixing A, there are many choices for…

Q: 2 Let A and B be two measurable setsa prove that ie BAIS measurable and mCB\A) =M CB)-mcA) %3D

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Q: Prove that S UT = SN T for all sets S and T using the rules for complements of sets.

A: rules of complements x∈A⇔x∉A¯

Q: An operation * on a set S is a relationship that assigns to some ordered pair of elements of S a…

A: We have to tell about the statement is whether true or 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.