The cartesian product \(A \times B \times C\) contain triplets of the form (a,b,c).
Thus, the elements belong to the set:
\(A \times B \times C = \left\{ {\left( {a,b,c} \right)a \in A \wedge b \in B \wedge c \in C} \right\}\)
Whereas, the cartesian product \(A \times B\) contain doublets of the form (a,b).
Thus, its elements belong to the set:
\(A \times B \times C = \left\{ {\left( {x,c} \right)x \in A \times B \wedge c \in C} \right\}\)
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And s the cartesian product \(\left( {A \times B} \right) \times C\) contain triplets of the form (x,c).
Thus, the elements belong to the set:
\(A \times B \times C = \left\{ {\left( {a,b,c} \right)a \in A \wedge b \in B \wedge c \in C} \right\}\)
Now, using Cartesian product , the product \(\left( {A \times B} \right) \times C\) is obtained as:
\(\left( {A \times B} \right) \times C = \left\{ {\left( {a,b} \right),c\left| {\left( {a \in A \wedge b \in B} \right) \wedge c \in C} \right.} \right\}\)
\(\left( {A \times B} \right) \times C = \left\{ {\left( {\left( {a,b} \right),c} \right)\left| {a \in A \wedge b \in B \wedge c \in C} \right.} \right\}\)
Thus, \(A \times B \times C\) contains triplets while \(\left( {A \times B} \right) \times C\) contains doublets.