Products

Given two sets \(A\) and \(B\), we take their binary Cartesian product \[(A × B)\] to be the set of pairs of elements \(⟨a, b⟩\), where \(a ∈ A\) and \(b ∈ B\). We call \(a\) the first component (or projection) of such a pair and \(b\) the second component. Given a pair \(p\), we will sometimes write ‘\(p_{1}\)’ to refer to its first component and ‘\(p_{2}\)’ to refer to its second component. In particular: \[⟨x, y⟩_{1} = x\] \[⟨x, y⟩_{2} = y\]

The following statement is also true: \[⟨x_{1}, x_{2}⟩ = x\] That is, if you take the first and second components of a pair and pair them back up, you haven’t done anything!

We can generalize binary products to \(n\)-ary products by considering the former to correspond to the special case where \(n = 2\). For the general case \[A₁ × ... × A_{n-1} × Aₙ\] we can encode this product as \[(...(A₁ × ... × A_{n-1}) × Aₙ)\] For example, \(A × B × C\) is just \(((A × B) × C)\); its members are, therefore, pairs \(⟨⟨a, b⟩, c⟩\), which is just how we can encode 3-tuples \(⟨a, b, c⟩\).