Formatting/notation isn’t important. Lots of options are fine. You can use what you prefer, and you can find problems with any notation.
Truth tables are useful when there are variables that could be true or false. If you evaluate an expression with 1’s and 0’s, no variables, then you don’t need a table.
Next you need the concept of defining logic operators in terms of other operators.
For example:
nand(x,y) = ?
Answer only using the following operators: not, and, or.
Max 30min.