Logical
sum of two variables |
Generalization
of the properties of the logical operations |
General
symbols of use (American representation) |
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Created it, 06/09/09
Update it, 06/09/13
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2. 8. - LOGICAL PRODUCT OF TWO VARIABLES
2. 8. 1. - EXAMPLE AND DEFINITION
Let us suppose now that one asks: Is Jacques at the school ? YES or NOT. There are again two possible answers.
But now, Paul AND Jacques are at the school ?
Let us summarize the various answers on the table (figure 22).
We see that when “Paul and Jacques are at the school” “Jacques is needed is at the school” AND “Paul is at the school.”
We have just defined the switching function AND.
We can write P = a . b which we will read P = a AND b.
2. 8. 2. - REPRESENTATION OF EULER OR VENN (figure 23).
If unit A is the unit for which the variable a = 1, and the unit B that for which the variable b = 1, P is the intersection of A and B, and represents the time during which Jacques and Paul are at the school, i.e. the whole of the values of P equal to 1.
2. 8. 3. - ELECTRIC ASSEMBLY (figure 24)
So that the lamp ignites, it is necessary to press on a AND b simultaneously. Up to that point, we drew the contacts (term used by the logicians) or switches (more general term) as on figure 24 (representation which is very much used).
Let us study now the representation of figure 25.
Positive logic makes correspond at the physical state contact closed the logical state 1, and at the physical state open contact the logical state 0. The switch is always represented at rest falling from its own weight : what is not new.
We noted in the treating chapter of information in the numerical systems that the physical phenomenon considered is often a tension as that of the numerical signal which we described with two levels high and low well defined.
We adopted the Anglo-Saxon terms High (H) which means high and Low (L) which means low because they are very often used by large manufacturers of components such Texas or in certain literatures. We will use this notation in certain examples in order to accustom you as of now to these two levels high and low.
So that the lamp S is lit, it is necessary a and b closed i.e. at the high level from where tables of figure 26.
2. 8. 4. - PROPERTIES OF THE LOGICAL PRODUCT
Let us associate a binary variable x with 0 and 1, itself or its complement.
Let us give an illustration of these logical associations by means of electrical contacts by using positive logical convention :
1°) x
. 0 = 0
(figure 27)
When one presses on button x, the variable passes to 1 but the power could never be on and S will be always extinct from where the truth table of figure 27.
2°) x .
1 = x (figure 28)
When one presses on button x, variable x only acts on the current, indeed, the contact always with 1 does not have an influence because it always lets pass the current from where the truth table of figure 28.
3°) x .
x = x (figure 29)
When one presses on two buttons x, the variable is to 1 and the two contacts are closed so that the lamp S ignites. S takes the value of x from where the truth table of figure 29.
4°) 
(figure 30)
When one presses on the button,
contact x is closed and the contact 
opens, the lamp being thus never fed, therefore always extinct, from where the
truth table of figure 30.
2. 9. - LOGICAL SUM
OF TWO VARIABLES
2. 9. 1. - EXAMPLE AND DEFINITION
Let us suppose now that two children of the village, Paul and Jacques go to the school of the close city. When there will be at least a child of the village in the bus of school bus service which led them there ?
Let us summarize the possibilities on the table of figure 31.
We see that there is a child of the village in the bus when Paul OR Jacques is in the bus, but not only when one or the other is inside, but also when they are both there.
This is why we say that the function S = f (a,b) is one OR INCLUSIVE (because it included the case where a and b are present at the same time).
In Boolean algebra (by typographical convenience), one notes the operation OR inclusive +.
Example :
a + b = 1 which is stated a OR b is worth 1.
Contrary to the algebra traditional, the sign + does not mean more but OR, indeed 1 + 1 = 1 in Boolean algebra!
We previously saw what is fundamental, that the values taken into account are states and not numbers.
2. 9. 2. - REPRESENTATION OF EULER OR VENN (figure 32)
That is to say the logical variables a and b. Let us draw in the reference frame Â, surface or the unit A inside whose variable a is to 1 and the unit B inside whose the variable b is to 1.
The unit in which the variables a or b are to 1, or logical sum, will be the formed surface of the meeting of the two preceding areas.
In Boolean algebra, we will use only the notation :
S = a + b which one reads S = a OR b
2. 9. 3. - ELECTRIC ASSEMBLY (figure 33)
We see that it will be enough to close a OR b so that the lamp S ignites (or both at the same time).
One can write S = a + b from where figure 34 :
2. 9. 4. - PROPERTIES OF THE LOGICAL SUM
Let us associate a binary variable x with 0 and 1, itself or its complement.
For that, we will give an illustration of this logical association by means of electrical contacts (positive logical convention).
1°) x + 0 = x (figure 35).
When one presses on button x, variable x passes to 1 and the current supplies the lamp S from where the truth table of figure 35.
2°) x + 1 = 1 (figure 36)
When one presses on button x, variable x passes to 1, but this does not have an effect because the lamp S remains lit permanently from where the truth table of figure 36.
3°) x + x = x (figure 37)
Two contacts x are mechanically dependant, they are closed simultaneously, we could replace figure 37 by figure 38 : the tables of truth are identical for S compared to x. One of contacts x can be removed.
When the power is not on by x
it passes by
and the lamp S is always supplied, from where the truth table of figure
39.
2. 10. - GENERALIZATION
OF THE PROPERTIES OF THE LOGICAL OPERATIONS
We can deduce from the properties seen previously : the summary table (figure 40).
1°) Commutation
One can write that a + b = b + a what is checked in the example of the table (figure 40) : 0 + 1 = 1 and 1 + 0 = 1 but also a . b = b . a what is checked in the same table (figure 40): indeed 0 . 1 = 0 and 1 . 0 = 0.
2°) Idempotence
For the logical sum one can write : a + a + a = a
For the logical product one can write : a . a . a = a
This property also rises from the table of figure 40.
3.a) Distributivité of the logical product compared to
the logical sum
One can easily check that :
a (b + c) = ab + ac
This property is called distributivity of the logical product compared to the logical sum; it is identical to the distributivity of the multiplication compared to the addition in the traditional algebra.
3.b) Representation of Euler
Figure 41 shows the 3 sets a, b, c.
Let us draw figure 42 the unit S = a (b + c) intersection of a and b + c.
Let us build figure 43 now, the ab sets and bc like their meeting S = ab + ac.
We see that surface S is the same one in both cases.
4.a) Absorption
Let us consider a + ab that one can write a (1 + b), but we know that b + 1 = 1, from where it is deduced that a + ab = a (1 + b) = a . 1 = a.
One calls the property a + ab = a property of absorption.
4.b) Representation of Euler
Let us consider the sets a and ab in the representation of Euler of figure 44.
We see easily that the unit a + ab is not other than a.
One can say that :
When a sum contains a term and one of its multiples, one can neglect the multiple.
5.a) Somme of a variable and a multiple of its
complement.
Let us consider S = a +
b.
By using the corollary of the property of absorption which is perfectly valid, one can write :
S = a + b
= (a + ab) + b
= a + ab + b
from where S = a + b (a +
)
however a +
= 1
from where S = a + b
a + b
= a + b
One will be able to say that when a logical sum is made up of the sum of a variable and a multiple of its complement, one can make disappear the complement.
5.b) Representation of Euler
(figure 45)
We see that b
is the zone corresponding to : b (hatchings blue and red) and
that a + b
is not other than a + b.
6.a) Distributivité of the logical sum compared to the
logical product.
Let us consider S = a + bc
The property of absorption makes it possible to write :
a = a + ab ; a = a + ac and like a + a = a
One can write a = a + ab + a + ac which becomes a = ab + ac + a
from where S = a + ab + ac + bc
Like a = a . a
S =aa + ac + ab + bc
S = a (a + c) + b (a + c)
from where S = (a + b) (a + c)
a + bc = (a + b) (a + c)
6.b) Representation of Euler
(figure 46-a and 46-b)
One easily highlights by means of the circles of Euler figure 46 the equality (a + bc) = (a + b) (a + c).
2. 11. - GENERAL SYMBOLS
OF USE (AMERICAN REPRESENTATION)
1.) Function YES
It will be materialized by an electric wire generally.
It could be materialized by the symbol of the buffer (figure 47). We will speak later about the buffer.
2.) Function NOT (figure
48)
The symbol of the function NOT is characterized by the addition of a bubble which shows that the exit of the logical operator is reversed.
3.) Function AND (figure 49)
4.) Function OR (figure
50)
It should be noted that these symbols are those of logical operators contained in the integrated circuits, for example, and although they require a power supply, this one is not represented.
We will continue to look further into this lesson on another page in order not to encumber this one.
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