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A one to one correspondence (one to one and onto function) from a set A to A is called a permutation on A. For any non-empty set A, let S(A) represents the set of all permutations on A. That is S(A) = {f:A → A\f is both one to one and onto}. Let * be an operation on S(A) defined by V f,g E S (A), f *g = fog. a. Is the operation binary? b. Determine whether the operation is associative and/or commutative. Prove your answers or give a counter example. c. Determine whether the operation has identities. d. Determine if every element in S(A) has inverse?

A one to one correspondence (one to one and onto function) from a set A to A is called a permutation on A. For any non-empty set A, let S(A) represents the set of all permutations on A. That is S(A) = {f:A → A\f is both one to one and onto}. Let * be an operation on S(A) defined by V f,g E S (A), f *g = fog. a. Is the operation binary? b. Determine whether the operation is associative and/or commutative. Prove your answers or give a counter example. c. Determine whether the operation has identities. d. Determine if every element in S(A) has inverse?

Advanced Engineering Mathematics
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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A one to one correspondence (one to one and onto function) from a set A to A is called a permutation on A. For any non-empty set A, let S(A) represents the set of all permutations on A. That is S(A) = {f:A → A\f is both one to one and onto}. Let * be an operation on S(A) defined by V f,g E S (A), f *g = fog. a. Is the operation binary? b. Determine whether the operation is associative and/or commutative. Prove your answers or give a counter example. c. Determine whether the operation has identities. d. Determine if every element in S(A) has inverse?
Transcribed Image Text:A one to one correspondence (one to one and onto function) from a set A to A is called a permutation on A. For any non-empty set A, let S(A) represents the set of all permutations on A. That is S(A) = {f:A → A\f is both one to one and onto}. Let * be an operation on S(A) defined by V f,g E S (A), f *g = fog. a. Is the operation binary? b. Determine whether the operation is associative and/or commutative. Prove your answers or give a counter example. c. Determine whether the operation has identities. d. Determine if every element in S(A) has inverse?
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