Hello everyone,I would like to ask is there anybody can solve these questions?
1.)Let A = {a, b}.Does this table define a semigroup and monoid on A?
* a b a a b b a b
2.)Let S = {a,b}. Write the operation table for the semigroup (P(S), ∪).
I really do not have ideas on how to solve them,that is why I ask here.
Please explain briefly on how to do the questions,thanks!
There are 16 possible "triple products". For example, 2 of them would be:
(a*a)*a and a*(a*a)
For us to have a semigroup, these would have to be equal no matter whether "a" or "b" was in any position.
From the table, we have:
a*a = a, so:
(a*a)*a = (a)*a = a*a = a while:
a*(a*a) = a*(a) = a*a = a. So for a product involving all a's, we have associativity.
Now YOU do the other 7 comparisons.