48 Prove that if f is strictly monotone on a set ESR , then f is one-to-one on E.

Q: 35. For each real number x, let f(x)=x². For any two real numbers a and b, define a -b provided that…

A:

Q: Give a proof of the following statement. Suppose A, B and C are sets, f is an injective function…

A: We have to prove the given statement.

Q: Let {E} be a countable collection of sets of real numbers. Then UEn ≤Σm* (En) n * m n

A: Definition of outer measure: The outer measure of a set E is defined as, m*(E)=inf∑j=1∞Qj, where…

Q: 9. Let A be a nonempty set. Prove that if f is a bijection on A and f o f = f. then f is the…

A: Let A be a nonempty set. f:A→A be a bijective function on A. Such that  f◦f = f (f◦f)(x) = ?   ; for…

Q: Show that, for any set A, there is exactly one map f from the empty set ∅ to A. When is f injective?…

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Q: 5. Let f : R → R. Prove that f is continuous on R + f-(H) is a closed set whenever H is a closed…

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Q: Suppose that f is a function from A to B, where A and B are finite sets with |A| = |B|. Show that f…

A: Given that f is a function from A to B, where A and B are finite sets with A=B. We have to show that…

Q: Define a function f: Z* to Z* by the rule: for each integer n, f(n) = the sum of the positive…

A:

Q: 4. Suppose X is a finite set and A, B C X. Show that if |A| + |B| > |X| + 3, then |AN B| 2 3. 5.…

A: We use induction. This clearly holds whenever n=1. Fix n∈N and suppose that every function from…

Q: A relation R is defined on the set of real numbers by xRy if and only if x-y is a multiple of 3.…

A:

Q: 1. Let f be defined on an interval I and let x1 and x2 be numbers on I. If f(x1) < f(x2) whenever x1…

A: Given that: f is deifned in I such that:   f(x1)&lt;f(x2) whenever x1&lt;x2

Q: 2. Answer the questions with Yes or No, and then explain why yes or why not. Justify your answer. a)…

A: The solution of given problem is given as follows :

Q: 7. Prove that if f and g are continuous then so are max(f, g) and min(ƒ, g).

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Q: show that A0 subset f-1 (f(A0)) and that equality holds if f is injective

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Q: Let f : R" → R be continuous. Prove or give a counter example: if O c R" is open, then f(O) is open.

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Q: a. Prove that Rf is an equivalence relation. b. Suppose that f is the squaring function defined by…

A: This is a problem of Relation.

Q: Consider functions f and g such that composite gof is defined and is one-one. Are f and g both…

A: Given f and g such that composite gof is defined and is one-one.

Q: | Suppose that f is O(g) and that g is O(h). Prove that f is O(h).

A:

Q: let G be 5 Such that

A: let  G be a group H , K be subgroup of G such that NGH=NGK claim- relate H , K

Q: Let A be a subset of Real numbers. Prove that A is Open iff  A Compliment (A') is a closed set.

A:

Q: Prove whether the following theorem is true or not : If A is any countably infinite set, B is any…

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Q: Let R be a reflexive relation on A. Show that R" is reflexive for all positive integers n.

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Q: ; Let 'L' and 'M' are two non-empty sets which are defined as; L = {x; -1 <x<1} = M and modulus…

A: Injective function

Q: Give an example of a function f. N –> N (N is the set of natural numbers) that is 1. 2. 3. 4. (note:…

A: Note: There are many subparts in question. So I will answer the first three subparts in the…

Q: Give a proof of the following statement. Suppose A, B and C are sets, f is a surjective function…

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Q: a. Let S = {0, 1}. Let W be the set of finite strings over the set S. Define a function f:W → set of…

A: (a) f(0) = length of '0'              = 1 [Ans]

Q: If f is a function having as its domain and range the set N of natural numbers, then f is bijective.

A: For the solution follow the next step.

Q: Prove Proposition 3.4. A monotone function that is defined on an interval is measurable.

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Q: Show that if f and g are continuous on [a, b] and if f(a) g (b), then there exists some ce ]a, b[…

A:

Q: 7. Find a counterexample or a proof for the following statement: If f: A- B and g : B- C are…

A:

Q: Suppose that f is holomorphic in an open set 2. Prove that in any one of the following cases: 2 3. f…

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Q: Show that the set {e^n: n ∈Z} is countable. (Hint: Use the function f (n) = e^n and show it is 1-1…

A:

Q: (i) Show that F(S) is an infinite set, and that f is injective but not surjective. (i) Is F(S)…

A: iiFS is an infinite set:First we note that S must be an infinite set as no of primes is infinitehere…

Q: Suppose f: A --> B is a function and g is the powerset of B to the powerset of A, while g(S) =…

A:

Q: Let f : X → Y, A¡ C X for all i e I and B; C Y for all j e J where I and J are some nonempty index…

A:

Q: Suppose f : N → N is an injective function on the natural numbers which, considered as a relation,…

A:

Q: Let f : X → Y, A¡ C X for all i e I and B; C Y for all j e J where I and J are some nonempty index…

A:

Q: 6. Find a counterexample or a proof for the following statement: If f: A- Band g: B- C are functions…

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Q: (ii) Let A be a non-empty set and f: AA be a function. Prove that if fof is injective, then / is

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Q: Let X = {1,2, 3}. Define the function f from P(X) to {0, 133as the power set. Which elements are not…

A:

Q: Let S be the set of all odd integers, and define a function f: Z → S as follows: f(n) = 2n + 1 for…

A:

Q: 5. Let X and Y be sets and let f: X→Y be onto. For all be Y, let As = f-¹({b}). (a) Prove that P =…

A: Given that X and Y are the sets , f : X→Y be onto function and ∀b∈Y, Ab=f-1b. To prove: (a) P is a…

Q: 7. Let A and B be sets, and let f: A → B and g: B → A be functions. Suppose that: • go f(a) = a for…

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Q: 1. Let X be a set with n elements. Compute the number of distinct functions f: X → X. How many of…

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Q: 27. Let F be the set of all finite sets. Is F an element of itself?

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Q: Given g, a function from A to B and f, a function from B to C. "If f and g are onto, then f • g is…

A: Given that g:A→B is onto. If y∈B, then there exists a pre-image x∈A. ⇒g(x)=y

Q: If B is countable and f : P(A) → B is a one to one (i.e., injective) function, then A must be…

A: We have to find is the statement is true or false.

Q: Let F and G be a non-empty families of sets with SG. Show that ng s UG.

A: Given: F and G are two non-empty family of sets with F⊆G, To show: ∩F⊆∪G

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.