(b) Let f: D→ E be a function, let I be an arbitrary index set, and let A; be a subset of Efor all i E I. Prove that(i)~² (₁)-0²²(4)=ΕΙΕΙ

Given Data: Prove that:i f-1∩i∈I Ai=∩i∈I f-1Ai

Answered: (b) Let ƒ : D → E be a function, let I…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Prove the following statements: (a) If f : A → B and g : B → C are one-to-one functions, then go f : A → C is one-to-one. (b) Use the definition to prove that for any sets A, B and C' if |A| < |C| and |C| < |B| then |A| < |B|.
Let n be a positive integer and let f : [0..n] → [0..n] be an injective function. Define the function g : [0..n] → Z as g(x) = n - (f(x))². Prove that is also injective.
Let us define a number a to be a fixed point of a function f if a = f(a). (For example, if f (x) = -x², then x = -1 is a fixed point because f(–1) = -1.) Prove that if f'(x) # 1 for all numbers x, then f has at most one fixed point.
Let f A
Let f be a function on an interval I and let ₁ and 2 be any two points in I. Then if f (x₁) > f (x₂) whenever ₁ < x2, then f is said to be on I.
Prove that f is an identity function of itself.
Let f: X → Y be a function and A, BY be subsets of the codomain. a. Is f-¹(AUB) = f-¹(A) u f¹(B)? Always, sometimes, or never? Explain. b. Is f-¹(An B) = f¹(A) nf-¹(B)? Always, sometimes, or never? Explain.
A function f : Z → Z is defined by f(n) = 2n + 2. (a) Determine f(E), where E is the set of even integers. (b) Determine f-1(D), where D = {6k : k E Z}.
Which of the following relation is a function? a) Z={(1,4), (2,3), (3,2). (2,4), (1,4), (1,5 )} b) s={(- 2,1) (- 1,2) (1,2) (2,1) (3,4). (5,7)} c) A={1,3). (2. – 5 ). (1.5). (- 3,1) (- 2,–2).} d) M = {(2.1) (5.1) (8,1) (11.1). (1.1) (2.5)}
3. Let S {1,2, 3, 4} and define functions f,g : S → S by f = {(1,3), (2,2), (3, 4), (4, 1)} and g = {(1, 4), (2, 3), (3, 1), (4, 2)}. Find (a) g-1 o f og (b) f oglog (c) f-1 og –1 -1ofog
2. Let f: R→R be defined by f(x) = 4x -x2. Let B be the range of f. (a) Find B. (b) Fins f-¹([0, ∞ )). (c) Find a set ASR such that fl, the restriction of f on A, is a one-to-one and onto function from A to B, and find (fl)
Prove that An is gooup (०री ४०spect tp the function. a Composition of

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

(b) Let ƒ : D → E be a function, let I be an arbitrary index set, and let A¿ be a subset of E for all i E I. Prove that (i) ++ (0₁)=0²²(4) f-1 Ai = n f¯¹ (A₁) iЄI iЄI