Not one-one, (c) not single-valued ?
I)(ii)(iii)
A B C DA B C DA B C D ↓↓↓
A + 0 0 +A 0 + 0 0A 0 0 0 0
B 0 0 + 0B 0 0 0 +B + 0 0 +
C + 0 0 0C + 0 0 0C 0 + 0 0
D 0 + 0 +D 0 0 + 0D 0 0 + 0
Ex. 3: Can a closed transformation have a matrix with (a) a row entirely of zeros?
B) a column of zeros ?
Ex. 4: Form the matrix of the transformation that has n' = 2n and the integers as
operands, making clear the distribution of the +’s. Do they he on a straight
line? Draw the graph of y = 2x; have the lines any resemblance?
Ex. 5: Take a pack of playing cards, shuffle them, and deal out sixteen cards face
upwards in a four-by-four square. Into a four-by-four matrix write + if the
Card in the corresponding place is black and o if it is red. Try some examples
And identify the type of each, as in Ex. 2.
Ex. 6: When there are two operands and the transformation is closed, how many
Different matrices are there?
Ex. 7: (Continued). How many are single-valued ?
The generation and properties of such a series must now be con-
Sidered.
Suppose the second transformation of S.2/3 (call it Alpha) has
Been used to turn an English message into code. Suppose the
Coded message to be again so encoded by Alpha— what effect will
This have ? The effect can be traced letter by letter. Thus at the first
Coding A became B, which, at the second coding, becomes C; so
Over the double procedure A has become C, or in the usual nota-
tion A → C. Similarly B → D; and so on to Y → A and Z → B.
Thus the double application of Alpha causes changes that are
Exactly the same as those produced by a single application of the
Transformation
↓
A B … Y Z
C D … A B
Thus, from each closed transformation we can obtain another
Closed transformation whose effect, if applied once, is identical
With the first one’s effect if applied twice. The second is said to be
The “square” of the first, and to be one of its “powers” (S.2/14). If
The first one was represented by T, the second will be represented
By T2; which is to be regarded for the moment as simply a clear
And convenient label for the new transformation.
Ex. 1: If A:
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a b c ↓ c c a' what is A2?
Ex. 2: Write down some identity transformation; what is its square?
Ex. 3: (See Ex. 2/4/3.) What is A2?
Ex. 4: What transformation is obtained when the transformation n' = n+ 1 is
Applied twice to the positive integers? Write the answer in abbreviated
form, as n' = . . . . (Hint: try writing the transformation out in full as in
S.2/4.)
Ex. 5: What transformation is obtained when the transformation n' = 7n is applied
Twice to the positive integers?
Ex. 6: If K is the transformation
↓ A B C
A
B
C
0
0
+
+
0
0
+
0
0
R E PEA TED C H A N GE
Power. The basic properties of the closed single-valued
Transformation have now been examined in so far as its single
Action is concerned, but such a transformation may be applied
More than once, generating a series of changes analogous to the
Series of changes that a dynamic system goes through when active.
16
What is K2? Give the result in matrix form. (Hint: try re-writing K in some
Other form and then convert back.)
Ex. 7: Try to apply the transformation W twice:
W: ↓ f g hg h k
17
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
The trial in the previous exercise will make clear the impor-
Tance of closure. An unclosed transformation such as W cannot be
Applied twice; for although it changes h to k, its effect on k is
Undefined, so it can go no farther. The unclosed transformation is
Thus like a machine that takes one step and then jams.
Elimination. When a transformation is given in abbreviated
arm, such as n' = n + 1, the result of its double application must be
Found, if only the methods described so far are used, by re-writing
He transformation to show every operand, performing the double
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Application, and then re-abbreviating. There is, however, a
Quicker method. To demonstrate and explain it, let us write out In
full he transformation T: n' = n + 1, on the positive integers, show-
Ing he results of its double application and, underneath, the gen-
Eral symbol for what lies above:
T: ↓ 1 2 3 … n …
2 3 4 … n' …T: ↓ 3 4 5 … n" …
n" is used as a natural symbol for the transform of n', just as n' is
The transform of n.
Now we are given that n' = n + 1. As we apply the same trans-
formation again it follows that n" must be I more than n". Thus
n" = n' + 1.
To specify the single transformation T 2 we want an equation
that will show directly what the transform n" is in terms of the
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