Sets (Pure Maths)

An real ordered pairs, (a, b), is given by the first number a and the second number b. For example, (1, 3), (3, 1), and (1, 1) are three different ordered pairs.
We now explain how ordered pairs of real numbers correspond to points in a plane R^2 . A system of rectangular coordinates in a plane is given by a horizontal and a vertical copy of the real line crossing at zero. The horizontal line is called the horizontal axis, or x-axis, while the vertical line is called the vertical axis, or y-axis. The point where the two axes meet is called the origin and corresponds to the ordered pair (0, 0). Now consider any point P in the plane. A vertical line through P will cross the x-axis at a real number x0, and a horizontal line through P will cross the y-axis at a real number y0. The ordered pair (x0, y0) obtained in this way corresponds to the point P. (See Figure 1) We sometimes call P the point (x0, y0) and sometimes write P(x0, y0). x0 is called the x-coordinate of P and y0 the y-coordinate of P.

圖片參考：http://www.vias.org/calculus/img/01_real_and_hyperreal_numbers-12.gif

Figure 1
Conversely, given an ordered pair (x0, y0) of real numbers there is a corresponding point P(x0, y0) in the plane. P(x0, y0) is the point of intersection of the vertical line crossing the x-axis at x0 and the horizontal line crossing the y-axis at y0. We have described a one-to-one correspondence between all points in the plane and all ordered pairs of real numbers.