Cartesian product

Description:

• Let $A$ and $B$ be sets.
• The Cartesian product of $A$ and $B$:
  • denoted by $A\times B$
  • is the set of all ordered tuples, each with pairs of elements $(a,b)$
    • where $a\in A$ and $b\in B$.
• Hence, $A\times B=\{(a,b)|a\in A\wedge b\in B\}$

Cartesian product for multiple sets:

• The Cartesian product of the set $A_1,A_2,...A_n$, denoted by $A_1\times A_2\times ...\times A_n$, is the set of ordered n-tuple $(a_1,a_2,...,a_n),$ where $a_i\in A_i$ for $i=1,2,...n$.
• In other words, A_1 × A_2 × . . . × A_n = \{(a_1, a_2, . . . , a_n) | a_i ∈ A_i$$ for i = 1, 2, … n}$