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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.1: The Geometry And Algebra Of Vectors

Problem 24EQ

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Given the sets G and H, can you prove that (G-H) x (H-G) is a subset of (GxH)-(HxG)

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Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
Prove that if f is a permutation on A, then (f1)1=f.
In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
In Exercises 1324, prove the statements concerning the relation on the set Z of all integers. If 0xy, then x2y2.
Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Find mappings f,g and h of a set A into itself such that fg=hg and fh. Find mappings f,g and h of a set A into itself such that fg=fh and gh.
23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
In Exercises , prove the statements concerning the relation on the set of all integers. 17. If and , then .
(4) Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C) = (A × B) \ (A × C). Prove that (A x B) n (C x D) = (ANC) × (BND).

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