Part 1 Suppose HC Z is a subset that satisfies (a) If x EH then –x E H (b) If x, y E H then x + y E H Show that the relation xRy ⇒ x - yE H is an equivalence relation. (Hint: first show that 0 € H) Unimportant Remark: We denote the set of equivalences classes Z/H and read it as 'Z mod H'. This is called the space of cosets in group theory. This problem works much more generally: replace Z by any group G, and then H is called a subgroup of G. Solution. Part 2 Let nZ := {nx : x € Z} C Z be the subset of all E multiples of n. Show that nZ satisfies conditions (a) and (b) from the above problem. What is the equivalence relation defined above in this case? What are the space of all cosets? Solution.
Part 1 Suppose HC Z is a subset that satisfies (a) If x EH then –x E H (b) If x, y E H then x + y E H Show that the relation xRy ⇒ x - yE H is an equivalence relation. (Hint: first show that 0 € H) Unimportant Remark: We denote the set of equivalences classes Z/H and read it as 'Z mod H'. This is called the space of cosets in group theory. This problem works much more generally: replace Z by any group G, and then H is called a subgroup of G. Solution. Part 2 Let nZ := {nx : x € Z} C Z be the subset of all E multiples of n. Show that nZ satisfies conditions (a) and (b) from the above problem. What is the equivalence relation defined above in this case? What are the space of all cosets? Solution.
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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Transcribed Image Text:Part 1
Suppose HC Z is a subset that satisfies (a) If x EH
then -x E H (b) If x, y E H then x + y E H Show that
the relation xRy ⇒ x - y E H is an equivalence
relation. (Hint: first show that 0 € H) Unimportant
Remark: We denote the set of equivalences classes
Z/H and read it as ʼZ mod H'. This is called the space
of cosets in group theory. This problem works much
more generally: replace Z by any group G, and then
H is called a subgroup of G. Solution.
Part 2
Let nZ := {nx :x E Z} C Z be the subset of all
multiples of n. Show that nZ
satisfies conditions (a) and (b) from the above
problem. What is the equivalence relation defined
above in this case? What are the space of all cosets?
Solution.
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