Solving Logical Combinations of Equations
When you give a list of equations to Solve, it assumes that you want all the equations to be satisfied simultaneously. It is also possible to give Solve more complicated logical combinations of equations.
When you use Solve, the final results you get are in the form of transformation rules. If you use Reduce or Eliminate, on the other hand, then your results are logical statements, which you can manipulate further.
The logical statements produced by Reduce can be thought of as representations of the solution set for your equations. The logical connectives &&, || and so on then correspond to operations on these sets.
|eqns1||eqns2||union of solution sets|
|eqns1&&eqns2||intersection of solution sets|
|!eqns||complement of a solution set|
|Implies[eqns1,eqns2]||the part of eqns1 that contains eqns2|
You may often find it convenient to use special notations for logical connectives, as discussed in "Operators".