Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![(1) Consider the function f: R\{-1} → R given by f(x) :
your claim.
I+:
I
(2) Consider the function f: NxN→Q given by f(x, y) =
=
31 +
your claim.
Find the image of f and prove
Find the image of f and prove
(3) Let F be a field. Without looking up the proofs from the book, or notes from class/TUT prove
that the additive and multiplicative inverses in a field are unique.
(4) Prove "cancellation" for fields (without looking up the proofs):
(a) For all z, y, z € F, if z + z = y + 2, then x = y.
(b) For all z, y, z € F \ {0}, if z-z = y. z, then z = y.
(5) Let F = {0, 1, a, b} be a field with 4 elements. Prove that if 1 + 1 = 0, then a + b = 1.
(6) Let F be a field and a € F. Prove that -(-2) = 1.
(7) Let F be a field. Prove that (-1). z = -x.
(8) Let P = "(VER) (y € R) (r+y> 0)".
Let Q = "(VER) (Vy E R) [(ry > T) V (zy <y)]".
(a) Is P true? Justify your answer.
(b) Is Q true? Justify your answer.
(c) Let R be any statement. Is (PAQ) ⇒ R true? Justify your answer.
(9) Let F be a field. Write each of the field axioms using logical symbols (and common notation
for sets, like €).
(10) Let E denote the set of even integers, and O denote the set of odd integers.
(a) Let P="Every integer is either even or odd."
Write P using logic symbols (and common notation for sets, like E, Z, N, Q etc).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facc2e209-8c76-4eaa-a6c2-2cf75e37f5c6%2F8cd4bb5a-462a-46c8-a71f-3e78e340bc0a%2F0ss0o2a_processed.png&w=3840&q=75)
Transcribed Image Text:(1) Consider the function f: R\{-1} → R given by f(x) :
your claim.
I+:
I
(2) Consider the function f: NxN→Q given by f(x, y) =
=
31 +
your claim.
Find the image of f and prove
Find the image of f and prove
(3) Let F be a field. Without looking up the proofs from the book, or notes from class/TUT prove
that the additive and multiplicative inverses in a field are unique.
(4) Prove "cancellation" for fields (without looking up the proofs):
(a) For all z, y, z € F, if z + z = y + 2, then x = y.
(b) For all z, y, z € F \ {0}, if z-z = y. z, then z = y.
(5) Let F = {0, 1, a, b} be a field with 4 elements. Prove that if 1 + 1 = 0, then a + b = 1.
(6) Let F be a field and a € F. Prove that -(-2) = 1.
(7) Let F be a field. Prove that (-1). z = -x.
(8) Let P = "(VER) (y € R) (r+y> 0)".
Let Q = "(VER) (Vy E R) [(ry > T) V (zy <y)]".
(a) Is P true? Justify your answer.
(b) Is Q true? Justify your answer.
(c) Let R be any statement. Is (PAQ) ⇒ R true? Justify your answer.
(9) Let F be a field. Write each of the field axioms using logical symbols (and common notation
for sets, like €).
(10) Let E denote the set of even integers, and O denote the set of odd integers.
(a) Let P="Every integer is either even or odd."
Write P using logic symbols (and common notation for sets, like E, Z, N, Q etc).
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