4. (a) Define f : R×R→R×R by f(x,y) = (2x – y, x – 2y). Prove that f is one-to-one. Describe f-1. What is ƒ-'(6,3)? Show that (f o f-1)(x, y) = (f-1 o f)(x, y). {a1, a2, ..., an} where n is a positive integer. Assume that a function (b) Let A f : A → A is onto. Prove that f is a bijection.
4. (a) Define f : R×R→R×R by f(x,y) = (2x – y, x – 2y). Prove that f is one-to-one. Describe f-1. What is ƒ-'(6,3)? Show that (f o f-1)(x, y) = (f-1 o f)(x, y). {a1, a2, ..., an} where n is a positive integer. Assume that a function (b) Let A f : A → A is onto. Prove that f is a bijection.
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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Transcribed Image Text:4. (a) Define f : RxR → R× R by f(x, y) = (2x – y, x - 2y). Prove that f is one-to-one.
Describe f-1. What is f-'(6,3)? Show that (fo f-1)(x,y) = (f-1 o f)(x, y).
(b) Let A
f : A → A is onto. Prove that f is a bijection.
{a1, a2, ..., an} where n is a positive integer. Assume that a function
Transcribed Image Text:(b) Let G be a simple graph with 20 vertices. Suppose that G has at most two com-
ponents, and every pair of distinct vertices u and v satisfies the inequality that
deg(u) + deg(v) > 19. Prove that G is connected.
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