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# Exercise 89 - Chapter 1

edited June 2018

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 $$c: T \twoheadrightarrow P$$.

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edited July 2018

The partitions on $$T$$ are

,,,,.

The images of these partitions under $$g^*$$ are

,,,,

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 $$T$$).

Comment Source:The partitions on \$$T\$$ are ,,,,. The images of these partitions under \$$g^*\$$ are ,,,, 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\$$).