Logical
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Electric
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Created it, 06/09/09
Update it, 06/0912
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1. - CONCEPTS ON THE SETS
1. 1. - SETS
One calls together, any collection of objects or beings taken randomly or having common properties.
Example :
the whole of the plants
the whole of the heavenly objects
the whole of the animals
the whole of the carnivores
the whole of the mammals
the whole of the integers.
In mathematics, one represents a whole by a surface.
1. 2. - INTERSECTION OF TWO SETS (figure 1)
One calls intersection of two sets A and B unit I composed of all the objects common to A and B.
The unit hatched I constitutes the intersection of sets A and B (figure 1).
In the mathematical set theory, one writes I = A n B and one states I equal A inter B.
In Boolean algebra, by typographical convenience, one writes :
I = A . B
That one states
I equal A AND B
Indeed, if for example A is the whole of the carnivores and B the whole of the mammals, I will be the whole of the mammalian AND carnivorous animals.
1. 3. - TOGETHER DISJOINS (figure 2)
One calls disjoined sets two sets A and B which do not have any common element.
It will be said that if for example A is the whole of the mammals and B the whole of the plants, that A and B are two disjoined sets.
There is no common point between the two.
1. 4. - MEETING OF TWO SETS (figure 3)
One calls meeting of two sets A and B, the unit R made up of the elements belonging to at least of sets A and B.
The unit hatched R constitutes the meeting of A and B. In the set theory one writes R = A u B and one states R equal A union B.
In Boolean algebra, by typographical convenience, one writes :
R = A + B
That one states R = A OR B
Indeed, if for example R is the whole of the alive beings, A the whole of the plants, and B the whole of the animals, one can say that the alive beings R are plants A OR animals B.
1. 5. - INCLUSION, SUBSET, LEFT (figure 4)
1. 5. 1. - EXAMPLE AND DEFINITION
If each element of a unit A belongs to the unit F, one can say that unit A is included or contained as a whole F.
One writes A Ì F. One states A is included in F.
Indeed, if for example F is the whole of the animals, and A the whole of the mammals, one will say that A is a subset of F, or that the whole of mammals A is only part of the whole of the animals F.
1. 5. 2. - WRITINGS EQUIVALENT TO A Ì F
a) - writing A . F = A is illustrated figure 5.
We can say, indeed, that the whole of the beings which are at the same time animals AND mammals is the whole of the mammals quite simply, that is to say A.
One can see that the intersection of A AND F is well A.
A n F = A or A . F = A
But we also see that A is well included in F.
b) - The writing A + F = F is illustrated figure 6.
One can say that A u F = F or A + F = F.
Unit A is well included in F.
This comes down to saying for example that a being, pertaining to the whole of mammals A, or pertaining to the whole of the animals F, belongs in any event to the whole of the animals F.
1. 6. - COMPLEMENT OF A WHOLE COMPARED TO ANOTHER (figure 7)
That is to say a unit A included in a unit E.
The whole of the elements belonging to E and not belonging to A is the complement of A compared to E.
One will note 
and one will state A bars the complement of A compared to E.
If E for example is the whole
of the animals, and A the whole of the carnivores (or animals eating of
the meat), one can say that the animals vegetarians (or not eating meat) belong
to the unit .
1. 7. - REFERENCE FRAME (figure 8)
Let us suppose that unit A (p) is the whole of the elements having the property p included in a unit Â.
The  unit is called reference frame.
If no element of unit A (p) has the wished property, one will call A (p) empty set.
The unit (p)
is the complement of A (p) compared to Â.
By convention, in Boolean algebra, the reference frame is indicated by 1 and the empty set by 0.
If the empty set is
and the reference frame  =
1, one can
write:
A +
= Â =
1
1. 8. - EXAMPLE OF REPRESENTATION OF BINARY NUMBERS (figure 9).
One calls binary digit a figure which can have only two values 1 or 0.
Let us consider the  reference frame consisted the portion of the plan P delimited by a closed curve. The plan is thus shared in two zones not recutting itself. The reference frame contains the binary logical variables taking values 1 or 0.
If there exists a, b, c, d, such as a = 1 for A pertaining to A, b = 1 for b Î B, c = 1 for c Î C and d = 1 for d Î D, one will say that a = 0, b = 0, c = 0 and d = 0 for a, b, c, d not belonging to A, B, C, D ; i.e. belonging to the empty set 0, (Î means pertaining to, Ï means not belonging to).
We can say that inside the reference frame there are nine numbered zones from 0 to 14 such as a, b, c, d take the values of figure 10.
We can see that all the combinations of a, b, c, d, do not exist, such as is drawn figure 9.
We will see thereafter that certain binary variables (terms defined in the following chapter) cannot take in an electric diagram, for example, all the existing values in consequence of technological impossibilities.
2. - LOGICAL ALGEBRA
2. 1. - CONCEPT Of STATE
“The state is the manner of being things.”
It will be said that a salad is green and that a tomato is red. But one will be able to never add a green salad and a red tomato.
On the other hand, one can say without being mistaken that there are two plants (from which the colors are different).
If we now consider a red bicycle and a red car, we can say that they are of red color.
We thus see that if operations can be carried out by taking account of states (here colors) they are not arithmetic operations, or algebraic traditional, because the result is not characterized by a number but by a state (here a color).
In the preceding examples two different characteristic colors were selected, the red and the green : our logic of reasoning is thus a logic in two states.
“It is necessary that a door is opened or closed.”
It is the same thing for an electrical contact, for which there are only two positions (or two states) : opened or closed.
An electromagnetic relay like those used in the telephone exchanges or self-switching is at rest (nonexcited) or with work (excited).
A shutter in a pneumatic control is opened or closed, it will be the same for a hydraulic valve.
A proposal such as Paul is at the school can have only two logical answers or states :
YES Paul is at the school
NOT Paul is not at the school.
2. 2. - BOOLEAN BINARY VARIABLE
In the example chosen previously, we can say : Is Paul at the school ? YES or NOT. Nothing prevents you from YES affecting the answer value 1 and the response NOT of value 0.
We can say that the variable “Paul is at the school” has two states : YES state 1, NOT state 0 (figure 11).
A Booléenne variable will be thus very quantity likely to take only two values : 1 or 0.
Let us study the duration of the day now :
When Paul is at the school, the days make 24 hours.
When Paul is not at the school do the days make 24 hours ? YES.
We can say that in our history, the duration of the day is a constant.
If we arbitrarily assign state 1 to the answer YES, the state of the day will be always 1.
We will call Booléenne constant any Booléenne quantity which always keeps the same value either 1, or 0.
2. 3. - FUNCTION OF A VARIABLE
When two Boolean variables, a and b, are bound by a relation, such as to a value of a corresponds a value of b, it is said that b is related to a and one writes :
b = f (a)
Example :
“If the weather is nice, I will walk.”
Here the variable “walk” is a function of the variable “good weather”. Indeed, if it rains I will not walk, but if the weather is nice I will go.
2. 4. - FUNCTION OF SEVERAL VARIABLES
To simplify, let us examine the case of a function of two variables.
A function will be known as of two variables when the variable c, for example, depends at the same time on the value of a variable a, as well as variable b. It is said that c is related to a and b and one writes c = f (a,b).
Example :
“If the weather is nice and if I am in better health, I will walk.”
The variable “walk” depends on the variable “good weather” and the variable “better health”. Because, it is necessary so that I will walk that the weather is good and that my health is better.
2. 5. - FUNCTION OF FUNCTION
If d is related to c, one can write d = F (c) but like c = f (a,b) one writes :
d = F [ f (a,b) ]
It is said that d is related to function of a and b.
Example :
“If I will walk, I will take along the dog.”
The variable “left the dog” is a function of the variable “walk” which, as we saw higher is a function of the variable “good weather” and of the variable “health”.
2. 6. - FUNCTION YES (figure 12)
2. 6. 1. - EXAMPLE AND DEFINITION
Let us suppose now, that when the weather is nice, Paul will play balloon and that when it rains it goes to the school.
Let us call respectively a and b the variable “good weather” and the variable “Paul plays balloon.”
If we YES affect the response of 1 and the response NOT of 0, we can build the table (figure 12) :
We see that the variable “Paul plays balloon” is to 1 when the variable “good weather” is to 1. It is to 0 when the variable “good weather” is 0.
One can write Paul plays balloon = f (good weather) and like the variable “Paul plays balloon” always takes the same value as the variable time, one says that the function f is a function YES.
One writes a = b.
2. 6. 2. - REPRESENTATION OF EULER OR VENN (figure 13)
It is said that two variables a and b are equal, when they are represented by the same points of the reference frame i.e. if contours which define them are confused.
2. 6. 3. - ELECTRIC ASSEMBLY
:
CONVENTIONS (figure 14)
The push rod switch of figure 14 is always on a general diagram represented at rest, i.e. the pushbutton which makes it operate not actuated. by convention, one represents the push rod so that it falls in position by its own weight (contact represented horizontal).
On figure 14, we see a normally-open contact, i.e. which is closed with work (when it is actuated). It is thus well opened at rest as on figure 14.
By convention also one makes correspond in positive logic :
at the physical state contact closed the logical state 1
at the physical state open contact the logical state 0
One names the “variable contact” : a,
because this variable is active to 1, i.e.
lets pass the current from the pile when one presses on the push rod : it
is a normally-open contact.
The lamp ignites (state 1) when switch a is closed (state 1).
The lamp is extinct (state 0) when switch a is open (state 0).
One defines by state “0” or state “1” the electric state of an element.
2. 6. 4. - TRUTH TABLE
Up to now, we studied logical proposals such as we of make each time we speak ; i.e. sentences.
Now, in the practical examples carried out with contacts, and illustrating each function, we make correspond in a physical state a logical state.
In the example of figure 14, we can call “a” variable of entry and “S” variable of exit or receiver. In general, a receiver will be an engine, a lamp or any ordered body. A variable of entry will be a contact or a transducer or sensor which can summarize itself with a contact.
This is why, the truth tables are summarized with the variables of affected entry and exit of the logical state corresponding. For figure 14, we obtain the truth table (figure 16).
We will use small letters for the variables of entry and capital letters for the variables of exit.
2. 7. - FUNCTION INVERSION, NOT, NOT
2. 7. 1. - EXAMPLE AND DEFINITION
Now let us examine the relation or function which binds the variable “good weather” to the variable “Paul is at the school” which we will call respectively a, b (figure 17).
We see that the variable “Paul is at the school” is to 0 when the variable “good weather” is to 1 and vice versa. We can thus say that the function f which binds the variable “Paul is at the school” with the variable “good weather” is the function inversion which one calls also complement.
We write b =
that we state b = a bar.
2. 7. 2. - REPRESENTATION OF EULER OR VENN (FIGURE 18)
That is to say a unit A included in a unit reference frame  of the binary numbers 0 or 1.
The whole of the elements belonging to  and not belonging to A, is called complement of A compared to Â.
The hatched unit (figure 18) is the complement of A compared to the  unit is B.
One can indicate the complement of A
by ,
one from of deduces whereas B =
.
As the  unit is the whole of the bits (one calls bit a binary digit of English “binary digit”) with 1 or 0, one can say that if the variable a Î A and if the variable b Î B.
if a = 0 ---------> b =
= 1
if a = 1 ---------> b =
= 0
2. 7. 3. - ELECTRIC ASSEMBLY : CONVENTIONS (figure 19)
The contacts being always represented at rest, (falling by their own weight), the contact represented figure 19 is always closed at rest : it will be called normally-closed contact.
We will keep the convention stated about figure 14 with knowing :
electric state of the contact : closed -----> logical state 1
electric state of the contact : opened
-----> logical state 0
We will call this convention, logic positive.
One calls in positive logic the
variable “normally-closed contact”
because this variable is a contact closed i.e. in a logical state 1
when it is not actuated i.e. when it is not active and one says whereas
= 1, the lamp S will then be lit and its logical state will be 1.
Contrary, when one supports on
the contact will open from where the logical state 0 : in this case
= 0 ; the lamp S will be then extinct and its logical state will be
0.
One can thus write S =
which one can summarize by the truth table (figure 20).
One defines by state “0” or state “1” the electric state of an element (figure 21).
We
advise you before further going comparing this figure with
figure 15 in order to include/understand the difference in notation well
between normally-open contact
and normally-closed
contact for which one calls
the variable .
We will continue this lesson on another page in order not to encumber this one.
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