User:IssaRice/Conjunction of subset statements versus Cartesian product subset statement
This is an exercise from Tao's Analysis I (exercise 3.5.6) and from Munkres's Topology (exercise 1.1.2(k)–(l)). It is an exercise that makes apparent some of the subtleties of the basic logic used in math.
The exercise is to consider the sets and the two statements:
- and ; and
and to ask whether one implies the other (or whether the two statements are equivalent).
Relationship between the statements
In general, (1) implies (2). In other words, if and , then .
If and are nonempty sets, then (1) and (2) are logically equivalent. In other words, and if and only if .
If and are both empty, then (1) and (2) are logically equivalent; in fact, they are both true.
Each statement written out in first-order logic
Erroneous general proof
Here is an erroneous general proof that (2) implies (1).
Exercise. Identify the error in the proof.
Expand to see solution:
Here is a counterexample that shows that in general, (2) does not imply (1). Let , , , and . Then
Thus (2) is true but (1) is false.