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Activity 2-2: Mapping a set to itself
Let’s take the natural numbers, 0, 1, 2, 3...
Think of a function f taking the natural numbers to
itself as follows;
Every odd number goes to its double.
Every number divisible by 4 goes to itself.
Every other even number goes to half itself.
What happens to the numbers above as f is repeated?
Which points go to themselves?
(We say they have period 1.)
Which points go around in a cycle?
(The period now is bigger than 1.)
Which periods are possible here?
We have here a bijection;
each number goes to a unique number,
and each number is ‘hit’ by a unique number.
We say the mapping is 1-1 (one-to-one.)
This means that the inverse function exists –
we can go backwards if we want to.
Task: what is the inverse function here?
Try to think of a different bijection,
where a different range of periods
are possible.
Now let’s think about another bijection for some infinite set S.
Let’s suppose one point goes to itself,
two points are in a cycle (or orbit) of length 2,
three are in a cycle of length 3 and so on.
These cycles generate an infinite sequence as follows:
There is clearly one point with period 1.
There are four with period 2, but we need to include
the point of period 1 here, as that returns to itself after
being transformed twice, so that makes five points.
What about period three?
Nine points plus the period 1 point again; 10 points.
Period 4 gives 16 points, plus the four period 2 points,
plus the period 1 point, so that’s 21.
So this situation has created the infinite sequence starting
1, 5, 10, 21, 26, 50, ...
(the rule generating this might be quite complicated!)
We can say that this sequence is realisable;
we can turn it into a picture of points with their orbits.
Now we can ask – which of the sequences
that we know and love from our everyday mathematical life
are realisable?
Task: try some sequences out
and see if they correspond
to orbit pictures like the ones above.
Fibonacci?
The sequence of squares?
The Lucas sequence has the same rule
as the Fibonacci sequence,
but starts 1, 3...
Worth a try...
If you create a sequence that is realisable,
check it on the Online Encyclopedia of Integer Sequences.
With thanks to:
Tom Ward
Carom is written by Jonny Griffiths, [email protected]