## Logical addition

* The logical addition M * of independent variables * X * m * (m * = 0, ..., M-1) is the operation, as a result of which a logical sum is obtained in the form of a logical function * Y * of the indicated independent variables, which is * equal to * 1 if at least one of the independent variables * X * m (OR X0, OR * X * v ..., OR * X * m, = 1) OR all variables are equal to one, and * is equal to * zero if all * X * m are equal to zero. Therefore, the logical addition is usually called the OR operation.

The truth table (Figure 3.4) shows the values of the logical function * Y * for four sets of input variables * X * y * X * 0. The logical addition is denoted by the symbol v, the entry * Y * = X, v * X * 0 is read: "or * X * v or X0 .

* Fig. 3.4. * ** The truth table of a logical addition element (OR), conditional graphic representation (a), schematic implementation ** * (b), * < strong> state of output Y = 0 (in) and

**Y =****1 (g)**

An element that implements logical addition is called an OR element, as well as a disjunctor. Its graphic designation and an example of a circuit implementation for * M = 2 * are shown in Fig. 3.4, * a, b. * The output signal is Y = 1 if at least one of the input signals X, or X0 is closed (Figure 3.4, * c). * And only for both open key (X, = Xd = 0) the signal * Y = * 1 (Figure 3.4, * r). *

## Logical multiplication

* Logical multiplication of the M * independent variables * X * m * (m * = 0, ..., * M -* * 1 ) is an operation that results in a product in the form of a logical function Y of these variables, which is* * equal to * 1 if all the variables * X * m (AND X0, AND X ,. .., ИХМ1) are equal to one, i.е. * Y * is zero if at least one of the variables * X * m is zero. Logical addition is also called an AND operation.

The truth table (Figure 3.5) shows the values of the logical function * Y * for * M * = 2. The operation of logical multiplication is indicated by the symbols •, a, & amp; To record a logical work, we usually use the first character (point) and even omit it. The logical product of two variables is written in the form: Y = X, • X0. The entry is read: and * X * x, and X0 .

The element by which logical multiplication is realized is called an AND element, or * a conjunctor. * Its graphic designation and an example of a circuit implementation for M = 2 are shown in Fig. 3.5, * a, 6. * In the conjunctor circuit, the signal Y = 0 if one of the series-connected transistor switches is open (Figure 3.5, * c). * For this it is necessary that although would be one of the signals

* Fig. 3.5. * ** The truth table of the multiplier (AND) logical element, the conditional graphic image ** * (a), *

**), exit state**

**Y****= 0**

*< strong> and*

**(in)**

**Y = i (g)** * X * v * X * 0 was equal to zero. And only for X, = * X * 0 * = * 1 the signal * Y = * 1, since all keys are in a closed state (Figure 3.5, d ).

Note that logical operations can be described using time diagrams (Figure 3.6). This type of representation is used for computer modeling of digital devices.

* Fig. 3.6. * ** Representation of basic logical operations in the form of time charts **

The basic logical operations are NOT, OR, and can be analytically described, and the logical elements are NOT, OR, AND - implement a combination device of any complexity. Let's consider the principles of creating expressions for describing the schemes of combinational devices.

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