Let A be a non-empty set and suppose that f : A → Rand g : A →R. Define f + g : A → R by ( f +g)(x) = f(x) + g(x) Provide an example that shows sup(f +g)(A) < sup f(A) + sup g(A) holds,with f (A) and g(A) both bounded above.

we take f(x)=-1 x∈Q∩A1 x∈Qc∩A and g(x)=2 x∈Q∩A-2 x∈Qc∩A(f+g)(x)=f(x)+g(x)=1 x∈Q∩A-1…

Let A be a non-empty set and suppose that f : A → Rand g : A → R. Define f + g : A → R by ( ƒ + g)(x) = f(x)+ g(x) Provide an example that shows sup(ƒ +g)(A) < sup f(A) + sup g(A) holds, with f (A) and g(A) both bounded above.

Let A be a non-empty set and suppose that f : A → Rand g : A → R. Define f + g : A → R by ( ƒ + g)(x) = f(x)+ g(x) Provide an example that shows sup(ƒ +g)(A) < sup f(A) + sup g(A) holds, with f (A) and g(A) both bounded above.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 24E: For any relation on the nonempty set, the inverse of is the relation defined by if and only if ....

See similar textbooks

Related questions

Q: A relation R defined on the set of real numbers is such that xRy and yRx = x = y: R is said to be A…

A: There are many types of relations Symmetric Asymmetric Reflexive Transitive

Q: Prove that a point x is a limit point of a set A if and only if every e-neighborhood of x intersects…

Q: (a) Let f(x) = px² + 2qx +r with p > 0. By considering the minimum, prove that f(x) > 0 for all real…

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

Q: Suppose that f is continuous on R and that f(x) = 0 whenever a is rational. Prove that f(x) = 0 for…

Q: Prove that if f is a continuous function with domain 0, 1 and X is the set of all x in 0, 1| such…

A: Suppose f is a continuous function with domain 0,1 If there is no 0≠x∈0,1 such that f is strictly…

Q: Prove that f(x) = 5x2 + 3x – 26 is θ(x2)

A: Given: f(x) = 5x2 + 3x – 26 is θ(x2)

Q: Theorem 4. Let X be a set, and let F: X Y be a l-1 and onto function. (a) F-lo F = Idx (b) FoF- =…

Q: Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function.…

A: Given f:K→X be a function, where K⊂ℝand X be any metric space with metric d. Let c∈ℝ and for some…

Q: Suppose a and c are real numbers, c > 0 and f is defined on -1,1] by fxª sin(]x|=) to if x # 0) if x…

Q: 2. Let a, b e R and let f be the function Z + Z defined by f(z) - ar + b. For which values of a and…

Q: Suppose that X and Y are topological spaces and f: X- Y. Show that f is continuous iff for each…

A: Definition: Let X and Y are topological spaces and f : X→Y be a function. We say that the function f…

Q: Suppose that f is a bijection and fog is defined. Prove: (i). g is an injection iff f o g is; (ii).…

A: Solution :-

Q: Prove that for all integers n > 2, f(x) = x/n is continuous on [0, 0).

Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

Q: 1. Let (X, M, µ) be a measure space, fn : X → [0, 1] a sequence of measurable func- tions. Show that…

Q: 2. !. (n!)². > Show that there is no holomorphic function f (z) on any open set containing 0 such…

A: To Prove: There is no holomorphic function fz on any open set containing 0 such that fn0=n!2.

Q: A point xo in D is said to be an isolated point of D if there is an r > 0 such that Dn (xo – r, x0…

Q: (b) Let F be a non-empty subset of R. Prove that if OF C F, then F is closed.

Q: a) Restate Definition 3.6, and check by elementary computations that the mapping g: R → R given by…

A: What is Lipschitz Function: A function is referred to as Lipschitz function if the norm of the…

Q: Let f be a function defined on an open interval I containing a. Prove that f'(a) exists and is equal…

Q: 1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B → R by…

A: we have to solve given problem:

Q: Let A be a subset of a space X and let x ∈ X. Then x ∈ A iff there exists a net in A (that is a net…

A: Given: Let us consider the given, A be a subset of a space X and take x ∈ X.

Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: For a bijective function f:(X,r1)→ (Y,12), one of the following statements is true: O f is open if…

Q: Let f : [a, b] → R be a continuous function and suppose that N is a natural number. If ¤1,..., IN…

A: It's a question from intermediate value theorem, so let's see how we can do it.

Q: Show that for any space A there does not exist an injection P(A) → A.

A: To prove that for any space A, there doesn't exist any injection PA→A Let us consider the contrary.…

Q: Find all values of m such that a(r) = {X – 1 if x m is continuous everywhere.

A: Questions from real analysis.

Q: 4. Let A C R" be both connected and sequentially compact. Prove that if F: A –→ R is continuous,…

A: Consider the given information.

Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

Q: 2. Let g : R → R be a bounded function and define f : R\ {0} → R by 1 f(x) = g(x)+ x2 Using only the…

Q: g) for any bounded

Q: Give an example (with justification) of a continuous function f : R→R such that for every non-empty…

Q: Let f be a differentiable real function defined in (a, b). Prove that f is convex if and only if f'…

A: A function f is convex in a domain a,b if for all 0≤λ≤1, fλx1+(1-λ)x2≤λfx1+1-λfx2 ∀x1,x2∈a,b. Also…

Q: Suppose that ci < c2 and that f takes on local maxima at ci and c2. Prove that if f is continuous on…

Q: Let X be a finite set and f: X → X. Prove that f is an injection iff f is a surjection. (Hint: If…

Q: If f and g are uniformly continuous and bounded, show that f.g is uniformly continuous. Give a…

Q: -B-Let f:( M, (Z).+-)-(M, (Z). +..) be a function define by such that g[]=[ EM₂(2). Is f is…

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: A relation < is defined on R by x < y if and only if there exists k E N such that x = y + 3k. Prove…

A: Partial order relation

Q: 5. Prove the following theorem from class: Theorem 1. Suppose that X and Y are nonempty finite sets…

A: In mathematics, a surjective function (also known as surjection, or onto function) is a function f…

Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: Suppose A is an uncountable set and B is a denumerable set. Suppose f:A→B is a function. Prove that…

Q: Suppose that f: C C is a holomorphic function such that there is a fixed kEN for which =1, for all.n…

Q: Let a and b be constant integers with a * 0, and let the mappingf: Z ----Z be defined by f(x) = ax +…

A: Given: f:ℤ→ℤ such that,f(x)=ax+b

Q: Let f,g be real-valued functions defined on a nonempty set X satisfying Range f and Range g are…

Q: Suppose a e R', fis a twice-differentiable real function on (a, ), and Mo, M1, M2 are the least…

A: "Since, the remaining conclusions are lengthy to prove, we will provide only the first proof. Kindly…

Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

Q: Given a closed set A, construct a continuously differentiable func- tion that has A as its set of…

Q: Suppose f : X → Y is a function. Prove that f is surjective if and only if for all sets Z and all…

Concept explainers

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”.

Question

Let A be a non-empty set and suppose that f : A → Rand g : A →R. Define f + g : A → R by ( f +
g)(x) = f(x) + g(x) Provide an example that shows sup(f +g)(A) < sup f(A) + sup g(A) holds,
with f (A) and g(A) both bounded above.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Sets$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .
18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove that. Prove that if.