Exercise 3.12. (1) Give an example of an ordered set X and an element a E X that has no predecessors. (2) Give an example of an ordered set X and an element a E X that has no successors.
Exercise 3.12. (1) Give an example of an ordered set X and an element a E X that has no predecessors. (2) Give an example of an ordered set X and an element a E X that has no successors.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please solve 3.12 in detail
![Definition 3.11. Let X be a nondegenerate set with linear order <. If there an element
a e X such that Vx E X,a < x, we say a is the least (or smallest) element of X. For
be X (b + a), we call [a, b) an initial segment of X. If there is an element w E X such
that Vx E X, x <w, we say w is the greatest (or largest) element of X. For be X (b+ w),
we call (b, w] a terminal segment of X.
Without commiting ourselves to the existence of least or greatest elements in X (o and
-00 are symbols used to denote rays, not elements of the set X), we define the following
four types of rays:
(1) For each a E X, the positive open ray from a is the set (a, 0) := {x € X | a < x}.
(2) For each a E X, the negative open ray from a is the set (-0, a) := {x E X | x < a}.
(3) For each a E X, the positive closed ray from a is the set [a, 00) := {x E X | a < x}.
TOPOLOGY NOTES
SPRING,
2021
3
(4) For each a E X, the negative closed ray from a is the set (-0, a] := {x € X | x < a}.
The positive open ray is sometimes called the set of successors of a; the negative open ray
is sometimes called the set of predecessors of a.
Exercise 3.12. (1) Give an example of an ordered set X and an element a E X that has
no predecessors. (2) Give an example of an ordered set X and an element a E X that has
no successors.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53e9ab80-64ca-4d50-9be9-589e5309635d%2F8561bb54-0ff6-49cf-b603-748c7f048abc%2Fybqb5de_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Definition 3.11. Let X be a nondegenerate set with linear order <. If there an element
a e X such that Vx E X,a < x, we say a is the least (or smallest) element of X. For
be X (b + a), we call [a, b) an initial segment of X. If there is an element w E X such
that Vx E X, x <w, we say w is the greatest (or largest) element of X. For be X (b+ w),
we call (b, w] a terminal segment of X.
Without commiting ourselves to the existence of least or greatest elements in X (o and
-00 are symbols used to denote rays, not elements of the set X), we define the following
four types of rays:
(1) For each a E X, the positive open ray from a is the set (a, 0) := {x € X | a < x}.
(2) For each a E X, the negative open ray from a is the set (-0, a) := {x E X | x < a}.
(3) For each a E X, the positive closed ray from a is the set [a, 00) := {x E X | a < x}.
TOPOLOGY NOTES
SPRING,
2021
3
(4) For each a E X, the negative closed ray from a is the set (-0, a] := {x € X | x < a}.
The positive open ray is sometimes called the set of successors of a; the negative open ray
is sometimes called the set of predecessors of a.
Exercise 3.12. (1) Give an example of an ordered set X and an element a E X that has
no predecessors. (2) Give an example of an ordered set X and an element a E X that has
no successors.
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