Let A, B, X, and Y be nonempty sets. Prove/disprove the following statements.a. Let f: X→Y. If A CX, BCY, and f-¹[B] = {x € X|f(x) = B}, then f[A] nB = f[Anf-¹[B]].b. Let f: A → B and g: B → A. If fog is the identity function is on B, then f is surjective.c. Let f: A→ B and g: B → A. If go f is the identity function i4 on A, then f is injective.

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Answered: Let A, B, X, and Y be nonempty sets.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Consider the set X = {a,b,c} and the function f: X→ X defined by the rules f (a) = b. f (b) = a and f (c) = b. Using the notation f(a) to indicate the function f composed with itself r times, write down explicitly the following functions: .fof . fofof • f(9) * f(624)
Answer c and d
Can you help me with this probelm?
5. Let f: CC be the function defined by f(z) = 25 for any z = C. Fill in the blanks in the blocks below, all labelled by capital-letter Roman numerals, with appropriate words so that they give respectively a proof for the statement (E) and a proof for the statement (F). (The 'underline' for each blank bears. no definite relation with the length of the answer for that blank.) (a) Here we prove the statement (E): (E) The function f is surjective. [We want to verify the statement (II) For this C, (IV) By definition, z E C. Note that f(z) = 2 = It follows that (VI) (b) Here we prove the statement (F): (F) The function f is not injective. (III) [We want to verify the statement (II) Note that zo, wo E C. Also note that We have (IV) and It follows that (VI) (III) (V) (1) 9 such that CC(cos(0) + sin(0)). (1) (V) Then f(zo) f(wo).
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 every a E A; fo g(b) : = b for every b E B. Prove that f is bijective.
if f is a one – to – one function from X into Y and {A, :s E S} is a collection of sets in P(X) then, n As =N f(A,) $ ES sES
For nonempty sets A, and B, let f: A →A and g: A → B be functions. Prove if the statements that are true, give counterexamples to disprove those that are false. "If f of is ono-to-one, then f is one-to-one."
For nonempty sets A, and B, let f: A → A and g: A → B be functions. Prove if the statements that are true, give counterexamples to disprove those that are false.
Let F be a "function" defined on the class of all sets. Then for each ordinal y, there is a unique function h such that 1. dom(h) = y 2. For each a
Let A and B be sets. Let f : A → B be a function. Let S ⊆ A be a subset. Prove that graph(f|S) = graph(f) ∩ (S × B).
1. For this question, use N as the universal set. Define f = {(X,Y) E P(N) × P(N) : XUY = N and XnY = 3, viewed as a relation from P(N) to P(N). Prove that f is a function P(N) → P(N).

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