1. Which of the following are equivalence relations? For those that are, describe the equivalence classes as best as you can. N a. On X = Zx yiff xy0 b. On X = Nay iff either (x > 100 and y> 100) or (<100 and y < 100) c. On X=Qxyiff x-yEN d. On X = Zx ~ yiff rye Z 2. For x, y ЄR, define x ~1 yiff x and y are both positive, both negative, or both 0. a. Show that b. Show that c. Show that is reflexive. is symmetric. is transitive. d. What is the equivalence class [1]? e. What is the equivalence class [-1]? f. What is the equivalence class 0? g. Show that every real number x is in exactly one of the classes [1], [-1] or [0].
1. Which of the following are equivalence relations? For those that are, describe the equivalence classes as best as you can. N a. On X = Zx yiff xy0 b. On X = Nay iff either (x > 100 and y> 100) or (<100 and y < 100) c. On X=Qxyiff x-yEN d. On X = Zx ~ yiff rye Z 2. For x, y ЄR, define x ~1 yiff x and y are both positive, both negative, or both 0. a. Show that b. Show that c. Show that is reflexive. is symmetric. is transitive. d. What is the equivalence class [1]? e. What is the equivalence class [-1]? f. What is the equivalence class 0? g. Show that every real number x is in exactly one of the classes [1], [-1] or [0].
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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![1. Which of the following are equivalence
relations? For those that are, describe the equivalence
classes as best as you can.
N
a. On X = Zx yiff xy0
b. On X = Nay iff either (x > 100 and
y> 100) or (<100 and y < 100)
c. On X=Qxyiff x-yEN
d. On X = Zx ~ yiff rye Z
2. For x, y ЄR, define x ~1 yiff x and y are both
positive, both negative, or both 0.
a. Show that
b. Show that
c. Show that
is reflexive.
is symmetric.
is transitive.
d. What is the equivalence class [1]?
e. What is the equivalence class [-1]?
f. What is the equivalence class 0?
g. Show that every real number x is in exactly
one of the classes [1], [-1] or [0].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c6bf976-c68c-4159-a094-199d2c95b159%2F1d1a28a1-68eb-456b-b70b-d44371e1828b%2F6y23y4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Which of the following are equivalence
relations? For those that are, describe the equivalence
classes as best as you can.
N
a. On X = Zx yiff xy0
b. On X = Nay iff either (x > 100 and
y> 100) or (<100 and y < 100)
c. On X=Qxyiff x-yEN
d. On X = Zx ~ yiff rye Z
2. For x, y ЄR, define x ~1 yiff x and y are both
positive, both negative, or both 0.
a. Show that
b. Show that
c. Show that
is reflexive.
is symmetric.
is transitive.
d. What is the equivalence class [1]?
e. What is the equivalence class [-1]?
f. What is the equivalence class 0?
g. Show that every real number x is in exactly
one of the classes [1], [-1] or [0].
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