Suppose that A x B = null set, where A and B are sets. What can you conclude?
By definition, . Now, when does it happen that any set of the form is empty? This happens when it is not the case that there exist x and y such that A(x,y), or, equivalently, when for all x and y, it is not the case that A(x,y). Can you state this more simply for this particular A?
Suppose that A x B = null set, where A and B are sets.It is given that both A and B are sets, though one or both can be empty.Can one be a set and another the null?
The easiest way to find the answer is to consider several cases. Suppose both A and B are nonempty. Then there exists an and a . By the definition of , then ; therefore, .
Suppose that A is empty, regardless of B. Then you can't find a single pair of and such that and . Therefore, . The same happens when B is empty.
The same can be shown more formally. I'll write for "there exists", for "and", for "or" and for negation. Then for any function and a predicate , iff . In our case, is , and is the function that returns a pair . So iff . This in turn is equivalent to , or .
Taking negation, iff iff iff .