Q. A: For any set A define the set -A=(y R13x A such that y=-x). Prove that if A c R is non-emptyand bounded then sup(-A) = -inf(A).Qi, B: State and Prove Monotone Convergence Theorem.Q. C. Prove that for any irrational number, there exists a sequence of rational numbers (x) converging to .A

1. A: Prove that if A⊂R is non-empty and bounded, then sup(−A)=−inf(A). Proof: 1. Definition of -A:…

Answered: Q. A: For any set A define the set…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 55E

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Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
30. Prove statement of Theorem : for all integers .
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.

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Q. A: For any set A define the set -A=(y R13x A such that y=-x). Prove that if A c R is non-empty and bounded then sup(-A) = -inf(A). Qi, B: State and Prove Monotone Convergence Theorem. Q. C. Prove that for any irrational number, there exists a sequence of rational numbers (x) converging to . A