# [SOLVED](The set of all integers) Z^2

#### katye333

##### New member
Hello all,

I'm having a lot of trouble when it comes to set notation.
For instance, what does (the set of all integers) $$\displaystyle Z^2$$ mean?
What values are contained in this set?

Sorry if I didn't use the MATH tags right.

#### Fantini

MHB Math Helper
Hello Katye! I assume you mean $\mathbb{Z}^2$. This is shorthand notation for $\mathbb{Z} \times \mathbb{Z}$, which is a cartesian product.

When we write $\mathbb{Z} \times \mathbb{Z}$ we mean a set with elements of the form $(a,b)$, where $a$ (or the first component) and $b$ belong to $\mathbb{Z}$. In general, whenever we have $A \times B$, with $A$ and $B$ sets, we write in set notation as follows:

$$A \times B = \{ (a,b) \in A \times B : a \in A, b \in B \}.$$

This is saying that the set $A \times B$ constructed from the sets $A$ and $B$ have elements denoted $(a,b)$ where the first component belongs to $A$ and the second belongs to $B$. Likely if you have more than three components: you'll always read each component as belonging to the corresponding set in order.

For example, try to tell in the following cases how are the elements in each set:

$$\mathbb{R} \times \mathbb{Z},$$ $$\mathbb{R} \times \mathbb{Q},$$ $$\mathbb{Z} \times \mathbb{Q} \times \mathbb{Z}.$$

Cheers!

#### katye333

##### New member
Hello Katye! I assume you mean $\mathbb{Z}^2$. This is shorthand notation for $\mathbb{Z} \times \mathbb{Z}$, which is a cartesian product.

When we write $\mathbb{Z} \times \mathbb{Z}$ we mean a set with elements of the form $(a,b)$, where $a$ (or the first component) and $b$ belong to $\mathbb{Z}$. In general, whenever we have $A \times B$, with $A$ and $B$ sets, we write in set notation as follows:

$$A \times B = \{ (a,b) \in A \times B : a \in A, b \in B \}.$$

This is saying that the set $A \times B$ constructed from the sets $A$ and $B$ have elements denoted $(a,b)$ where the first component belongs to $A$ and the second belongs to $B$. Likely if you have more than three components: you'll always read each component as belonging to the corresponding set in order.

For example, try to tell in the following cases how are the elements in each set:

$$\mathbb{R} \times \mathbb{Z},$$ $$\mathbb{R} \times \mathbb{Q},$$ $$\mathbb{Z} \times \mathbb{Q} \times \mathbb{Z}.$$

Cheers!
Thank you for responding!

So, for the first one of yours:
A would be $\mathbb{R}$ and B would be in $\mathbb{Z}$?

Now if I have a function defined as

Cheers!