3.1 Examples and Basic Properties

1.
a. Not an additive group; only 0 has an inverse.
c. It is an additive abelian group (as in Example 4), unity is img, multiplication is associative (it is function composition). But h img (f + g) = h img f + h img g fails: h img (f + g)(x) = h[f(x) + g(x)] while

img

These may not be equal. Note that (f + g) img h = f img h + g img h does hold.
3.
a. Clearly img and img. If and are in S then ...

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