I am really struggling with this problem, and so are my friends in the same course. Please refer to PDF for the tables.
Any help or guidance is appreciated!

Given a mathematical system M=({0,a,b,c},#, &) where the two operators # and

& are defined in the following two subtables.

1. Verify whether the system is a Boolean algebra.

2. List the complements of elements 0, a, b, and c if the system is a Boolean algebra.



I have started a table that my professor suggested to follow on the pdf, but I have no idea how to start it.

Another hint was that:

I found the following link is quite useful.

http://www.ctp.bilkent.edu.tr/~yavuz/BOOLEEAN.html

In table 1.(b), you can find something like very similar to "0" in table 1.(a).

Now you can say following:

(1) Something # Something' = 0 or a or b or c,

(2) Something & Something' = 0 or a or b or c.

From table 1.(a) and 1.(b), you can find pairs that satisfy your (1) and (2) always.

 

The first step is to make sure you understand what is required for something to be "a Boolean algebra".

So what, in YOUR words, is required?

 

Following the commutative, distributive, and identity laws is required I believe.

 

Look here, particularly at the definition and examples section. The reading is probably a bit difficult because they use higher-level math jargon, but you can probably get quite a bit of the gist of it.

https://en.wikipedia.org/wiki/Boolean_algebra_(structure)

 

The laws that the operators must follow are only part of the story. The properties of the elements are also important.