How to solve these questions?

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?

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.(Worried)

Please explain briefly on how to do the questions,thanks! (Happy)

Re: How to solve these questions?

Quote:

Originally Posted by

**xenosaga111** 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?

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.

What are the definitions involved here? How would they be applied here?

Please show some work.

Re: How to solve these questions?

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.

Re: How to solve these questions?

Are these all the possible combinations for a and b?

aaa

aab

aba

baa

abb

bba

bab

bbb

Re: How to solve these questions?

Actually,what do you mean by definition??Could you please show me some examples??

Sorry for double post.