Question: Let f : A B, g. B C. Prove that (a) if g o f: A C is onto, then g is

Let f : A → B, g. B → C. Prove that (a) if g o f: A → C is onto, then g is onto; and (b) if g o f: A → C is one-to-one, then f is one-to-one.

Step by Step Solution

★★★★★

3.44 Rating (170 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

a If c C there is an element a A ... View full answer

blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Document Format (1 attachment)

Word file Icon

954-M-L-A-L-S (7726).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Let f : A and g : B C and define g o f : A C by (g o f)(x) := g(f(x)). a) Show that if f, g are 1-1 (respectively, onto), then g o f is 1-1 (respectively, onto). b) Prove that if f is 1-1 from A...

Q:

Let E be a Jordan region in Rn. a) Prove that E and are Jordan regions. b) Prove that Vol(Eo) = Vol() = Vol(E). c) Prove that Vol(E) > 0 if and only if E 0. d) Let f: [a, b] R be continuous on [a,...

Q:

Let f : [a, b] R be C1 on [a, b] with f(t) 0 for t [a, b]. Prove that the explicit curve x = f-1(y), as y runs from f(a) to f(b), is orientation equivalent to the explicit curve y = f(x), as x...

Q:

Let f : A > B and g: h. : B > C. Prove that if f is onto and g 0 f : l1. 0 f then 9 : '1. Let f: g : A > B and h. : B > C. Prove that. if I1 is onetoono and h. o f : h o 9 then f = 9- Lot f : A > B...

Q:

Prove the following statements. (i) Let f: A B and g: B-C be two functions, If g of is onto, then g is onto. (ii) Let A, B, and C be three sets, then An (B-C)=(ANB)-(ANC). (iii) (AUB)-B=A-(ANB).

Q:

M AT H 1 2 0 Differential and Integral Calculus Fa l l 2 0 1 6 - P r o b l e m S e t # 1 1 Due Wednesday, November 30. Write your name and student number very clearly in the upper right hand corner...

Q:

Prove that composition of functions is associative. In other words, prove that, for functions f : A B, g : B C, and h : C D, ((h g) f)(x) = (h (g f))(x), for all x A.

Q:

Prove Theorem 5.9. If f: A -> B, g: B -> C are invertible functions, then g o f: A -> C is invertible and (g 0 f)-1 = f-1 g-1.

Q:

3. (20 marks) Let f : [a, b] - R be integrable. Prove there there exists a point c E [a, b] such that f(t) at = f(t) dt = f (t) dt. a (Hint: Use F(:) = f (t) dt for I E [a, b].) a

Q:

Given the matrices and vectors below, perform the operations indicated in a to d. (4 points each for a and d; 3 points each for e and f.) 0 0 Let A = o 0 1 2 2 1 4 B = . y = [1 0 0 1], and z = - N 0...

Q:

(a) Compared with a quantum of red light , a quantum of violet light has (1) more, (2) the same amount of, (3) less energy. Why? (b) Determine the ratio of the photon energy associated with violet...

Q:

Brody Company makes industrial cleaning solvents. Various chemicals, detergent, and water are mixed together and then bottled in 10-gallon drums. Brody provided the following information for last...

Q:

Which of the following is CORRECT regarding charitable gifts during life? A . When donating services to a charitable organization, the fair market value of the services provided and the unreimbursed...

Q:

In 1879, economist Henry George published Progress and Poverty, which became one of the best-selling books of the nineteenth century. In this book, George argued that all existing taxes should be...

Q:

You are constructing a future value spreadsheet. Users will be asked to enter the periodic investment in cell A3, the interest rate as an equivalent decimal in cell A4, the time in years in cell A5,...

Q:

Albert Einstein said that compound interest was ". . . the most powerful thing I have ever witnessed." Work through the following exercises to discover a pattern Einstein discovered which is now...

Q:

Robbie opens an account at a local bank by depositing $100. The account pays 2.4% interest, compounded weekly. He deposits $100 every week for three years. a. How much is in the account after three...

Q:

In addition to price - weighted and value - weighted indexes, an equally weighted index is one in which the index value is computed fro the average rate of return of the stocks comprising the index....

Q:

Which of the following statements regarding a 15-year (180-month) $125,000, fixed-rate mortgage is CORRECT? (Ignore taxes and transactions costs.) 1. The remaining balance after three years will be...

Q:

An investor is considering an investment that will pay $ 2 , 2 8 0 at the end of each year for the next 1 0 years. He expects to earn a return of 1 2 percent on his investment, compounded annually....