Transformation that is single-valued but not one-one will be
Referred to as many-one.
Ex. 1: The operands are the ten digits 0, 1, … 9; the transform is the third decimal
digit of log10 (n + 4). (For instance, if the operand is 3, we find in succession,
7, log107, 0.8451, and 5; so 3 → 5.) Is the transformation one-one or many-
One? (Hint: find the transforms of 0, 1, and so on in succession; use four-fig-
Ure tables.)
Ex. 5: If the operands are all the numbers (fractional included) between O and I,
and n' = 1/2 n, is the transformation closed? (Hint: try some representative
Values for n: 1/2, 3/4, 1/4, 0.01, 0.99; try till you become sure of the answer.)
Ex. 6: (Continued) With the same operands, is the transformation closed if n' =
1/(n + 1)?
The identity. An important transformation, apt to be dis-
Missed by the beginner as a nullity, is the identical transforma-
Tion, in which no change occurs, in which each transform is the
Same as its operand. If the operands are all different it is necessar-
Ily one-one. An example is f in Ex. 2/6/2. In condensed notation
n' = n.
Ex. 1: At the opening of a shop’s cash register, the transformation to be made on
Its contained money is, in some machines, shown by a flag. What flag shows
At the identical transformation ?
Ex. 2: In cricket, the runs made during an over transform the side’s score from
One value to another. Each distinct number of runs defines a distinct trans-
Formation: thus if eight runs are scored in the over, the transformation is
specified by n' = n + 8. What is the cricketer’s name for the identical trans-
Formation ?
The transformations mentioned so far have all been charac-
Terised by being “single-valued”. A transformation is single-val-
Ued if it converts each operand to only one transform. (Other
Types are also possible and important, as will be seen in S.9/2 and
Thus the transformation
↓
↓
A
B or D
A B C D
B A A D
BC
A B or C
D
D
Is single-valued; but the transformation
Is not single-valued.
Of the single-valued transformations, a type of some impor-
Tance in special cases is that which is one-one. In this case the
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Transforms are all different from one another. Thus not only does
Each operand give a unique transform (from the single-valued-
Ness) but each transform indicates (inversely) a unique operand.
Such a transformation is
Representation by matrix. All these transformations can be
Represented in a single schema, which shows clearly their mutual
Relations. (The method will become particularly useful in Chapter
And subsequently.)
Write the operands in a horizontal row, and the possible trans-
Forms in a column below and to the left, so that they form two
sides of a rectangle. Given a particular transformation, put a “+”
At the intersection of a row and column if the operand at the head
Of the column is transformed to the element at the left-hand side;
Otherwise insert a zero. Thus the transformation
A B C ↓ A C C
Would be shown as
ABC ↓
A+00
B000
C0++
The arrow at the top left corner serves to show the direction of the
Transitions. Thus every transformation can be shown as a matrix.
15
↓
A B C D E F G H
F H K L G J E M
This example is one-one but not closed.
On the other hand, the transformation of Ex. 2/6/2(e) is not one-
One, for the transform “1” does not indicate a unique operand. A
14
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
If the transformation is large, dots can be used in the matrix if
Their meaning is unambiguous. Thus the matrix of the transforma-
tion in which n' = n + 2, and in which the operands are the positive
Integers from 1 onwards, could be shown as
↓
1
2
3
4
5
…
100000…
2…00000
|
|
|
3…+0000
4…0+000
5…00+00
…………………
(The symbols in the main diagonal, from the top left-hand corner,
Have been given in bold type to make clear the positional relations.)
Ex. 1: How are the +’s distributed in the matrix of an identical transformation?
Ex. 2: Of the three transformations, which is (a) one-one, (b) single-valued but
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