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give an example of an associative system ( S,*) with identity for which cancellation laws fail.
Solution
The composition on B^B satisfies neither right nor left cancellation laws:
for example:
a) f : R R f(x) = x^2
g : R R g(x) = x g = idR
h : R R h(x) = x
then
f g(x) = x^2 , f h(x) = (x)^2 = x^2.
So f g = f h, but g does not equal to h

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give an example of an associative system ( S,) with identity for wh.pdf

  • 1. give an example of an associative system ( S,*) with identity for which cancellation laws fail. Solution The composition on B^B satisfies neither right nor left cancellation laws: for example: a) f : R R f(x) = x^2 g : R R g(x) = x g = idR h : R R h(x) = x then f g(x) = x^2 , f h(x) = (x)^2 = x^2. So f g = f h, but g does not equal to h