Introduction
- We have special symbols denoting the set of all integers and the set of all real numbers: Z and R respectively.
- These can be modified to denote the set of all ordered pairs:
- Z^{2} denotes the set of all ordered pairs of integers.
- R^{2} denotes the set of all ordered pairs of real numbers.
The set of all ordered pairs of integers
- Z^{2} contains every possible pair of integers, e.g.
- We can define Z^{2} in terms of Z using set builder notation:
LaTeX:
\mathbb{Z}^2 = \{ (z_1, z_2)\colon z_1, z_2 \in \mathbb{Z} \}
The set of all ordered pairs of real numbers
- R^{2} contains every possible pair of real numbers, e.g.
- We can define R^{2} in terms of R using set builder notation:
LaTeX:
\mathbb{R}^2 = \{ (r_1, r_2)\colon r_1, r_2 \in \mathbb{R} \}
Higher dimensions
- R and Z can also be extended to higher dimensions:
- To represent the set of all triplets of integers, you can use a superscript of 3: