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.

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    $\begingroup$ The usual definition of Cartesian product doesn't require that $A,B\neq \emptyset$. For example: en.wikipedia.org/wiki/Cartesian_product $\endgroup$
    – Micapps
    Jan 1, 2016 at 12:25
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    $\begingroup$ Regarding the tags: I don’t see how this is discrete mathematics and not elementary set theory. $\endgroup$ Jan 1, 2016 at 12:53
  • $\begingroup$ @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. $\endgroup$
    – buzzee
    Jan 1, 2016 at 12:56
  • $\begingroup$ @Jendrik Stetzner That chapter is included in other discrete mathematics books. Elementary set theory doesn't cover the cartesian product. $\endgroup$
    – buzzee
    Jan 1, 2016 at 12:58
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    $\begingroup$ @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. $\endgroup$ Jan 1, 2016 at 13:02

1 Answer 1


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.


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