UA-AUG-AUMAT229 Pre-read

Section 7.1 Pre-read

Definition 7.1.1. Bijective correspondence.

A correspondence between two collections so that each object in each collection is matched with one and only one object in the other collection.

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Example 7.1.2. Letters and numbers.

The correspondence

\begin{aligned} a & \leftrightarrow 1, & b & \leftrightarrow 2, & \dots, & z & \leftrightarrow 26 \end{aligned}

is bijective if we consider it as a correspondence between the (lowercase) alphabet and the collection consisting of the first twenty-six positive integers.

However, this correspondence would not be bijective if we considered it as a correspondence between the (lowercase) alphabet and the collection consisting of the first thirty positive integers, because in that case there would be some objects (numbers) in the second collection that are not matched up with any objects (letters) in the first collection.

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Example 7.1.3. Letters and numbers again.

The correspondence

\begin{aligned} 1 & \leftrightarrow a, & 2 & \leftrightarrow b, & \dots, & 26 & \leftrightarrow z, \\ 27 & \leftrightarrow a, & 28 & \leftrightarrow b, & \dots, & 52 & \leftrightarrow z \end{aligned}

is not bijective because some objects (letters) in the second collection have been matched up with more than one object (numbers) in the first collection.

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As you have already learned if you have taken AUMAT 250, bijective correspondences between infinite sets can do funny things.

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Example 7.1.4. Integers versus positive integers.

It would seem as if it should not be possible to set up a bijective correspondence between the collection of positive integers and the collection of (non-zero) integers, both positive and negative, because the second of those collections has “twice as many” objects as the first, So we would be forced into the same kind of situation as Example 7.1.3, where objects in one collection would have two matches in the other collection.

But if we get creative, the infinity of “ $\dots$ ” lets us get away with it:

1 \leftrightarrow 1, 2 \leftrightarrow - 1, 3 \leftrightarrow 2, 4 \leftrightarrow - 2, \dots .

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If we view a bijective correspondence as unidirectional instead of bidirectional, then really it is a function or “map” between the two collections, where objects in one collection are considered as inputs and objects in the other collection are considered as outputs. For example, we could rewrite the correspondence from Example 7.1.2 in function notation as

\begin{aligned} f (a) & = 1, & f (b) & = 2, & \dots, & f (z) & = 26 . \end{aligned}

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Function notation doesn't actually change the correspondence to a unidirectional one, it just temporarily “forgets” that it is really bidirectional. But we can always recover the bidirectionality by way of the inverse function:

\begin{aligned} f^{- 1} (1) & = a, & f^{- 1} (2) & = b, & \dots, & f^{- 1} (26) & = z . \end{aligned}

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And since a bijective correspondence must be bidirectional, what direction we initially choose if we decide to realize the correspondence as a function doesn't really matter; we could just as easily have started with

\begin{aligned} g (1) & = a, & g (2) & = b, & \dots, & g (26) & = a, \end{aligned}

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and then we would have inverse function

\begin{aligned} g^{- 1} (a) & = 1, & g^{- 1} (b) & = 2, & \dots, & g^{- 1} (z) & = 26 . \end{aligned}

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Definition 7.1.5. Operation-preserving map.

If we have a function $φ$ where the inputs are objects from one group and the outputs are objects from a second group, then we say that $φ$ is an operation-preserving map if

\begin{matrix} (✶) & φ (x y) = φ (x) φ (y) \end{matrix}

is always true, for each pair of inputs $x, y$ in the first group. That is, a mapping between groups is operation-preserving if the respective group operations can be performed either before or after applying the function, and the result will be the same either way.

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Warning 7.1.6.

In equality (✶) defining operation-preserving, there are (potentially) two different group operations involved: on the left the group operation of the “input” group is performed first and then the result is fed as the input to the function, and on the right the two inputs are fed into the function separately and then the group operation of the “output” group is applied to those two function outputs.

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Definition 7.1.7. Isomorphism.

An operation-preserving bijective correspondence between two groups.

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Definition 7.1.8. Isomorphic groups.

A pair of groups for which there exists at least one example of an isomorphism between them.

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Remark 7.1.9.

When two groups are isomorphic there might exist many different examples of isomorphisms between them.