4°)     (Figure 30)

When the button is pressed, the contact x closes and the contact opens, the lamp never being fed, therefore always off, hence the truth table of Figure 30.

2. 9. - LOGICAL SUM OF TWO VARIABLES

2. 9. 1. - EXAMPLE AND DEFINITION

Suppose now that two children from the village, Paul and Jacques go to school in the neighboring town. When will there be at least one child from the village on the school bus that takes them there ?

Let's summarize the possibilities on the table in Figure 31.

We see that there is a child from the village on the bus when Paul OR Jacques is on the bus, but not only when one or the other is in, but also when there are both.

This is why we say that the function S = f (a, b) is an INCLUSIVE OR (because it includes the case where a and b are present at the same time).

In Boolean algebra (for typographic convenience), we note the operation OR inclusive +.

Example :

a + b = 1 which is stated a OU b is 1.

Unlike traditional algebra, the sign + does not mean more but OR, indeed 1 + 1 = 1 in Boolean algebra !

We have seen previously what is fundamental, that the values taken into account are states and not numbers.

2. 9. 2. - REPRESENTATION OF EULER OR VENN    (Figure 32)

Let the logical variables a and b. Let's draw in the frame of reference Â, the surface or the set A within which the variable a is at 1 and the set B inside which the variable b is at 1.

The set in which the variables a or b are at 1, or logical sum, will be the formed surface of the union of the two preceding regions.

 

In Boolean algebra we will only use notation :

S = a + b  that we read  S = a OR b

2. 9. 3. - ELECTRICAL ASSEMBLY   (figure 33)

We see that it will be enough to close a OR b so that the lamp S lights up (or both at the same time).

We can write S = a + b from which Figure 34 :

2.9.4. - PROPERTIES OF THE LOGIC SUM

Let's associate a binary variable x with 0 and 1, itself or its complement.

For this, we will give an illustration of this logical association by means of electrical contacts (positive logical convention).

      1°)  x + 0 = x   (Figure 35).

When the x button is pressed, the variable x becomes 1 and the current supplies the lamp S, hence the truth table of Figure 35.

      2°)  x + 1 = 1   (Figure 36)

When the x button is pressed, the variable x becomes 1, but this has no effect because the lamp S remains permanently lit, hence the truth table in Figure 36.

      3°)  x + x = x   (Figure 37)

The two contacts x are mechanically linked, they close simultaneously, we could replace Figure 37 by Figure 38 : the tables of truth are identical for S with respect to x. One of the x contacts can be deleted.

When the current does not pass through x it passes through and the lamp S is always fed, hence the truth table of Figure 39.

2. 10. - GENERALIZATION OF THE PROPERTIES OF LOGICAL OPERATIONS

We can deduce from the properties seen previously : the summary table (Figure 40).

      1°)  Commutativity

We can write that a + b = b + a what is verified in the example of the table (Figure 40) : 0 + 1 = 1 and 1 + 0 = 1 but also a . b = b . a what is true in the same table (Figure 40) : indeed 0 . 1 = 0 and 1 . 0 = 0.

      2°)  idempotency

For the logical sum we can write : a + a + a = a

For the logical product we can write : a . a . a = a

This property also flows from the table in Figure 40.

      3. a)  Distributivity of the logical product with respect to the logical sum

We can easily check that :

a (b + c) = ab + ac

This property is called the distributivity of the logical product with respect to the logical sum ; it is identical to the distributivity of multiplication over addition in traditional algebra.

      3. b)  Representation of Euler 

Figure 41 shows the 3 sets a, b, c.

Draws Figure 42 the set S = a (b + c) intersection of a and b + c.

Let's build now Figure 43, the sets ab and bc and their meeting S = ab + ac.

We see that the surface S is the same in both cases.

      4. a)  Absorption

Consider a + ab that we can write a (1 + b), but we know that b + 1 = 1, from which we deduce that a + ab = a (1 + b) = a . 1 = a.

The property a + ab = a absorption property is called.

     4. b)  Representation of Euler

Consider the sets a and ab in the Euler representation of Figure 44.

We can easily see that the set a + ab is none other than a.

We can say that :

When a sum contains a term and one of its multiples, we can neglect the multiple.

      5. a)  Sum of a variable and a multiple of its complement.

Consider S = a + b.

By using the corollary of the absorption property which is perfectly valid, one can write :

S = a + b = (a + ab) + b = a + ab + b

from where   S = a + b (a + )

or  a + = 1

from where  S = a + b

a + b = a + b

We can say that when a logical sum is composed of the sum of a variable and a multiple of its complement, we can make the complement disappear.

      5. b)  Representation of Euler   (Figure 45)

We see that b is the area corresponding to : b (blue and red hatches) and a + b is nothing else that a + b.

      6. a)  Distributivity of the logical sum with respect to the logical product.

Consider S = a + bc

The absorption property makes it possible to write :

a = a + ab   ;   a = a + ac and like a + a = a

We can write a = a + ab + a + ac who becomes a = ab + ac + a

from where   S = a + ab + ac + bc

As 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   (Figures 46-a et 46-b)

It is easy to highlight by means of circles of Euler Figure 46 the equality (a + bc) = (a + b) (a + c).

2. 11. - GENERAL USE SYMBOLS (AMERICAN REPRESENTATION)

      1.)  Function YES

It will be materialized by a wire most often.

It can be materialized by the symbol of the buffer (Figure 47). We will talk about the buffer later.

      2.)  Function NO   (Figure 48)

The symbol of the function NO is characterized by the addition of a bubble which shows that the output of the logical operator is inverted.

      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, it is not represented.

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