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There are five partitions possible on a set with three elements, say \( T = {12, 3, 4} \).

**Example 1.86**.
Let \( S = {1, 2, 3, 4} \), \( T = {12, 3, 4} \), and \( g: S \rightarrow T \) by \( g(1) = g(2) = 12 , g(3) = 3, \text{ and } g(4) = 4 \).

Using the same \( S \) and \( g: S \rightarrow T \) as in Example 1.76, determine the partition \( g^*(c) \) on \( S \) for each of the five partitions \( c: T \twoheadrightarrow P \).

## Comments

The partitions on \(T\) are

[1234],[12][34],[123][4],[124][3],[12][3][4].

The images of these partitions under \(g^*\) are

[1234],[12][34],[123][4],[124][3],[12][3][4]

where now \(1\) and \(2\) are distinct elements of \(S\), though they always appear in the pulled-back partitions together (because \(g\) maps them to the same element of \(T\)).

`The partitions on \\(T\\) are [1234],[12][34],[123][4],[124][3],[12][3][4]. The images of these partitions under \\(g^*\\) are [1234],[12][34],[123][4],[124][3],[12][3][4] where now \\(1\\) and \\(2\\) are distinct elements of \\(S\\), though they always appear in the pulled-back partitions together (because \\(g\\) maps them to the same element of \\(T\\)).`