Relations

Cartesian product of three sets:
If A, B and C are three sets, then A x B xC=  {(a, b, c): a   A, b  B, c  C}
Ordered triplet is also called 3 – tuple
In general if A_­1, A₂,……, A_n are n sets, then (a₁, a₂,……. a_n) is called an – n tuple, where a_{1\( \in \)} A₁,I = 1, 2,….n and the set of all such n – tuples is called the Cartesian product of A₁, A₂….A_n and is denoted by      A₁ x A₂…….  A_n
In symbolic form
A₁ x A₂x …….  xA_n= \( \{ (a_1 ,a_2 ,.......a_n ):a_i \in A_i ,\,\,1 \leqslant i \leqslant n\} \)
Example 3: If A = { - 1, 1}, Find A x A x A.
Sol: A x A = (-1, 1) x (-1, 1) = {(-1, -1),(-1, 1),(1, -1),(1, 1)}
A x A x A = {(-1, -1), (-1, 1), (1, -1),(1, 1)} x (-1, 1)
= {(-1, -1, -1), (-1, -1, 1), (-1, 1, -1),(-1, 1, 1), (1, -1, -1),(1, -1, 1),(1, 1, -1), (1, 1, 1)}