1.3.6: Isomorphism

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An isomorphism is a bijection that preserves the structure of the sets it relates, where structure is a matter of the relationships that obtain between the elements of the sets. Consider the following two sets \(X=\{1,2,3\}\) and \(Y=\{4,5,6\}\). These sets are both structured by the relations successor, less than, and greater than. An isomorphism between the two sets is a bijection that preserves those structures. So a bijective function \(f \colon X \to Y\) is an isomorphism if, \(i<j\) iff \(f(i)<f(j)\), \(i>j\) iff \(f(i)>f(j)\), and \(j\) is the successor of \(i\) iff \(f(j)\) is the successor of \(f(i)\).

Definition \(\PageIndex{1}\): Isomorphism

Let \(U\) be the pair \(\langle X, R\rangle\) and \(V\) be the pair \(\langle Y, S\rangle\) such that \(X\) and \(Y\) are sets and \(R\) and \(S\) are relations on \(X\) and \(Y\) respectively. A bijection \(f\) from \(X\) to \(Y\) is an isomorphism from \(U\) to \(V\) iff it preserves the relational structure, that is, for any \(x_{1}\) and \(x_{2}\) in \(X\), \(\tuple{x_1,x_2} \in R\) iff \(\tuple{f(x_1),f(x_2)} \in S\).

Example \(\PageIndex{1}\)

Consider the following two sets \(X=\{1,2,3\}\) and \(Y=\{4,5,6\}\), and the relations less than and greater than. The function \(f\colon X \to Y\) where \(f(x) = 7-x\) is an isomorphism between \(\tuple{X,<}\) and \(\tuple{Y,>}\).