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Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x]

Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x]

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x] <x+1, where x is a real number. (Check all that apply.) Check All That Apply We can write the real number x as [x]+e, where is a real number satisfying 0 sr<1. Since E=x-[x], we have 0 sx-[x]<1. The first two inequalities x-1 < [x] and [x] ≤x follow algebraically. Since t = x-[x], we have 0 sx-[x]<1. The first two inequalities x-1s [x] and [x] <x follow algebraically, For the other two inequalities, we can write x= [x]-e, where again 0 se<1. Using 0 [x]-x<1, we can write xs [xland [x] <x+1.It shows that x-1 < [x] sxs [x] <x+1.
Transcribed Image Text:Identify the correct steps involved in proving that x-1 < [x] ≤x≤ [x] <x+1, where x is a real number. (Check all that apply.) Check All That Apply We can write the real number x as [x]+e, where is a real number satisfying 0 sr<1. Since E=x-[x], we have 0 sx-[x]<1. The first two inequalities x-1 < [x] and [x] ≤x follow algebraically. Since t = x-[x], we have 0 sx-[x]<1. The first two inequalities x-1s [x] and [x] <x follow algebraically, For the other two inequalities, we can write x= [x]-e, where again 0 se<1. Using 0 [x]-x<1, we can write xs [xland [x] <x+1.It shows that x-1 < [x] sxs [x] <x+1.
Let /be a function from A to B and S be a subset of 8. Identify the correct steps involved in showing that f-¹ (S) = f-¹ (S). (Check all that apply) Check All That Apply We know that S=B-S and d f-¹ (S) = A-1-¹ (s). We know that S=A-S and f-¹ (S) B-f f-¹ (S)=B-f-¹ (s). Let x Ef (S). This means that f(x) ES or equivalently that f(x) @S. Therefore, by the definition of inverse image, xer¹(5): so, x Ef-¹ (S). Therefore, by the definition of inverse image, xer¹(% 50, x Ef-¹ (S). Assume that x Ef-¹ (S). Then, x(S). This means that ) S It shows that f(x) ES. Therefore, by the definition of inverse image. x Ef-¹ (S).
Transcribed Image Text:Let /be a function from A to B and S be a subset of 8. Identify the correct steps involved in showing that f-¹ (S) = f-¹ (S). (Check all that apply) Check All That Apply We know that S=B-S and d f-¹ (S) = A-1-¹ (s). We know that S=A-S and f-¹ (S) B-f f-¹ (S)=B-f-¹ (s). Let x Ef (S). This means that f(x) ES or equivalently that f(x) @S. Therefore, by the definition of inverse image, xer¹(5): so, x Ef-¹ (S). Therefore, by the definition of inverse image, xer¹(% 50, x Ef-¹ (S). Assume that x Ef-¹ (S). Then, x(S). This means that ) S It shows that f(x) ES. Therefore, by the definition of inverse image. x Ef-¹ (S).
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