We can consider the case when they are rather smaller. When this
Case is known with certainty we can consider what happens when
They are smaller still. We can progress in this way, each step being
Well established, until we perceive the trend; then we can say what
Is the limit as the difference tends to zero. This, in fact, is the
Method that the mathematician always does use if he wants to be
Really sure of what happens when the changes are continuous.
Thus, consideration of the case in which all differences are
Finite loses nothing, it gives a clear and simple foundation; and it
Can always be converted to the continuous form if that is desired.
The subject is taken up again in S.3/3.
Next, a few words that will have to be used repeatedly. Con-
Sider the simple example in which, under the influence of sun-
Shine, pale skin changes to dark skin. Something, the pale skin, is
Acted on by a factor, the sunshine, and is changed to dark skin.
That which is acted on, the pale skin, will be called the operand,
The factor will be called the operator, and what the operand is
Changed to will be called the transform. The change that occurs,
Which we can represent unambiguously by
pale skin → dark skin
Is the transition.
The transition is specified by the two states and the indication
Of which changed to which.
T R A NS FO RM AT ION
The single transition is, however, too simple. Experience has
Shown that if the concept of “change” is to be useful it must be
Enlarged to the case in which the operator can act on more than
One operand, inducing a characteristic transition in each. Thus the
Operator “exposure to sunshine” will induce a number of transi-
Tions, among which are:
cold soil → warm soil
unexposed photographic plate → exposed plate
coloured pigment → bleached pigment
Such a set of transitions, on a set of operands, is a transformation.
Another example of a transformation is given by the simple
Coding that turns each letter of a message to the one that follows
It in the alphabet, Z being turned to A; so CAT would become
DBU. The transformation is defined by the table:
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A → B
B → C
…
Y → Z
Z → A
Notice that the transformation is defined, not by any reference to
What it “really” is, nor by reference to any physical cause of the
Change, but by the giving of a set of operands and a statement of
What each is changed to. The transformation is concerned with
What happens, not with why it happens. Similarly, though we may
Sometimes know something of the operator as a thing in itself (as
We know something of sunlight), this knowledge is often not
Essential; what we must know is how it acts on the operands; that
Is, we must know the transformation that it effects.
For convenience of printing, such a transformation can also be
Expressed thus:
A B…Y Z
B C…Z A
We shall use this form as standard.
Closure. When an operator acts on a set of operands it may
Happen that the set of transforms obtained contains no element
That is not already present in the set of operands, i.e. the transfor-
Mation creates no new element. Thus, in the transformation
A B…Y Z
B C…Z A
Every element in the lower line occurs also in the upper. When this
Occurs, the set of operands is closed under the transformation. The
Property of “closure”, is a relation between a transformation and
A particular set of operands; if either is altered the closure may
Alter.
It will be noticed that the test for closure is made, not by refer-
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Ence to whatever may be the cause of the transformation but by
Reference of the details of the transformation itself. It can there-
Fore be applied even when we know nothing of the cause respon-
Sible for the changes.
11
↓
↓
10
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
C H A NG E
Ex. 1: If the operands are the positive integers 1, 2, 3, and 4, and the operator is
Add three to it”, the transformation is:
1 2 3 4 ↓ 4 5 6 7
Is it closed ?
Ex. 2. The operands are those English letters that have Greek equivalents (i.e.
Excluding j, q, etc.), and the operator is “turn each English letter to its Greek
Equivalent”. Is the transformation closed ?
Ex. 3: Are the following transformations closed or not:
a b c df g p qB: ↓ A: ↓ a a a ag f q p
f g pD: ↓ f gg f qg f
Ex. 4: Write down, in the form of Ex. 3, a transformation that has only one oper-
And and is closed.
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