"nonempty" in the definition of the cartesian product

Definition 1: Let $A$ and $B$ be any nonempty two sets. The set of all ordered pairs $(x,y)$ is the Cartesian product of A and B, and is denoted by $A \times B$. In symbols $$A \times B = \{(x,y) : x \in A \wedge y \in B\}.$$

In the definition of the cartesian product, the cartesian product can't be made from two empty sets. The two sets should be nonempty. But in the following question the empty set $\emptyset$ makes up the two cartesian products, $A \times \emptyset$ and $\emptyset \times A$. I strongly feel that the the word nonempty should be removed from Definition 1. What do you think of Definition 1? Is it a complete definition for the cartesian product?

Question: Let A be any set. Find $A \times \emptyset$ and $\emptyset \times A$.

Solution: Since $A \times \emptyset$ is set of all ordered pairs $(a,b)$ such that $a \in A$ and $b \in \emptyset$​, and since the empty set $\emptyset$ contains no elements there is no $b \in \emptyset$; therefore $A × \emptyset = \emptyset$. Similarly, $\emptyset \times A = \emptyset$.

Source: In the chapter about Relations and Functions in the book Set Theory by You-Feng Lin, Shwu-Yeng T.Lin.

• The usual definition of Cartesian product doesn't require that $A,B\neq \emptyset$. For example: en.wikipedia.org/wiki/Cartesian_product Jan 1, 2016 at 12:25
• Regarding the tags: I don’t see how this is discrete mathematics and not elementary set theory. Jan 1, 2016 at 12:53
• @Micapps Yes, that was exactly I was thinking. I also checked the cartesian product definition in other books and websites and found that whether the sets should be empty or nonempty is not mentioned. Jan 1, 2016 at 12:56
• @Jendrik Stetzner That chapter is included in other discrete mathematics books. Elementary set theory doesn't cover the cartesian product. Jan 1, 2016 at 12:58
• @buzzee I am pretty sure that the cartesian product of two sets is a prime example of elementary set theory. It can be found in pretty much every text which covers some basic set theory, independent of the specific topic of the text. Jan 1, 2016 at 13:02

1 Answer

You are right.

In Set Theory with Applications, by You-Feng Lin and Shwu-Yeng T.Lin (2nd revised ed., 1981), page 62, we have the correct definition:

Definition 1. Let $$A$$ and $$B$$ be two sets. The set of all ordered pairs $$(x, y)$$, with $$x \in A$$ and $$y \in B$$, is called the Cartesian product of $$A$$ and $$B$$, and is denoted by $$A \times B$$. In symbols

$$A \times B = \{ (x,y) : x \in A ∧ y \in B \}.$$

Presumibely, your edition was subsequently corrected.