Question

数学简单证明 英语

Let A,B,C, and D be nonempty sets. Then A×B = C×D if and only if A=C and B=D.
(a) Prove this statement.
(b) One of the two implications does not require the sets to be nonempty. Which one?
(c) If we do not require the sets to bo nonempty, then the statement is false. Give examples of sets A,B,C and D to show the necessity of the assumption that the sets be nonempty.

Answer 1

(a) if A=C and B=D, of course AxB=CxD.
if AxB=CxD and A,B,C,D non-empty. If A is not equal to C, let's say A has an element a that does not belong to C. Since B and D are non-empty, there must be a pair with element a in AxC, but that pair does not belong to CxD since element a does not belong to C in the first place. So A must be equal to C. Likewise B must be equal to D.

(b) if A=C and B=D, then AxB=CxD.

(c) if A is empty, AxB is empty, if C is empty, CxD is also empty. So B can differ from D, but AxB still equals CxD.