Let $A$, $B$, and $C$ be sets where $A=\{1,2\}$, $B=\{0,1\}$, and $C=\{0,4\}$. What would be the Cartesian product of these $3$ sets for $A\times B\times C$?
I would think you would first determine $A\times B = \{(1,0), (1,1), (2,0), (2,1)\}$. Leading to $A\times B\times C=\{((1,0),0), ((1,1),0), ((2,0),0), ((2,1),0),((1,0),4), ((1,1),4), ((2,0),4), ((2,1),4)\}$.
I could also get rid of the parenthesis in part 1 to give $A\times B\times C=\{(1,0,0), (1,1,0), (2,0,0), (2,1,0),(1,0,4), (1,1,4), (2,0,4), (2,1,4)\}$.
Would part $1$ be correct, part $2$ be correct, or both part $1$ and part $2$ be correct? Any help would be greatly appreciated.
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$\begingroup$Assuming all you really have a definition of is an ordered pair, in terms of which longer tuples are defined, then technically you'll want something like 1; they'll be of the form $\langle\langle a,b\rangle,c\rangle$ or $\langle a,\langle b,c\rangle\rangle$. If this were part of some coursework, your instructor might want this level of technical correctness, or for you to at least know which of these you "really mean" if you write it as in 2.
Practically speaking, whether $\langle a,b,c\rangle$ is defined as either of the two pairs above or in some other manner is almost never important. All that will matter is that it's a set such that $\langle a,b,c\rangle=\langle d,e,f\rangle\Rightarrow a=d\wedge b=e\wedge c=f$, and we know how to find such sets if it becomes important to clarify.
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