Ordered Pair And Cartesian Product

Table of Contents

Ordered Pair

There are two elements in a pair. (Earth, Mars), {Maths, Physics}, (4,5), {a,b} are some examples of pair. Two pairs such as {Maths, Physics} and {Physics, Maths} have the same elements. Similarly, (4, 5) and (5, 4) also have the same elements. But in each case, the elements are distinct. If we consider a case in which the pair is (3, 3), then the pair has two identical elements. Two types of brackets are used here;

  1. Curly brackets { } or braces are used to denote sets. In sets, elements do not have the order of their occurence i.e. the elements can occur anywhere within { }. But, the repeatation of a same element is not allowed in sets.
  2. If the elements have the order in which they occur and an element can be repeated, then parentheses ( , ) with a comma between the elements are used. Such pair of elements is said to be ordered.

An ordered pair is a pair having one element as the first and the other as the second. An ordered pair having a as the first element and b as the second element is denoted by (a, b).

An ordered pair (a, b) is not the same as the ordered pair (b, a) unless the two elements are identical. Thus, (4, 5) is different from (5, 4); but (3, 3) is the same as (3, 3). Two ordered pairs (a, b) and (c, d) are equal if and only if a=c and b=d. Thus, two ordered pairs (2, 3) and (2, 3) are equal, but the ordered pairs (2, 3) and (3, 2) are not equal.

Cartesian Product

Let A and B be two given sets. Then, the set of all ordered pairs (a, b) such that a∈A and b∈B is called the Cartesian Product of A and B, and is denoted by A×B (read as A cross B). In the set builder notation, \[A×B=\{(a, b): a∈A \text{ and } b∈B\}\] Let A={1, 2, 3} and B={4, 5}. Then,

A×B={(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}

and, B×A={(4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3)}

The Cartesian Product of A={H, T} with itself, also known as the Cartesian Product on A is,

A×A={(H, H), (H, T), (T, H), (T, T)}

  • In general, A×B≠B×A.
  • If m is the number of elements in A and n is the number of elements in B, then the number of elements in A×B or B×A is mn.
  • If R is the set of real numbers, then the Cartesian Product of R on R i.e. R×R or R² is the set {(x,y):x∈R and y∈R}. This Cartesian Product is represented by the entire Cartesian Coordinate Plane.

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