@zareen said in MTH633 Assignment 1 Solution and Discussion:

Write down the elements of group under composition

Test 0: Is the operation associative? If not, then you’re done. 𝑇(𝑆)

is not a group. If the operation is associative, then proceed to…

Test 1: Does the set 𝑇(𝑆)

contains an identity? If you have a candidate function in 𝑇(𝑆), then what equations must it satisfy? Does it? If not, then 𝑇(𝑆)

is not a group. If you do have an identity, then proceed to…

Test 2: Does every element of 𝑇(𝑆)

have an inverse? Given an arbitrary function in 𝑓∈𝑇(𝑆), can you write down its inverse 𝑓−1∈𝑇(𝑆)? What equation must 𝑓 and 𝑓−1 satisfy? (Hint: you need the identity function.) If any function fails to have an inverse, then 𝑇(𝑆)

is not a group. If every function does have an inverse, then…

Congratulations! Your set 𝑇(𝑆)

is a group.

Reff