Logical addition, Logical multiplication - Informatics

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 .

The truth table of the logical addition element (OR), the conditional graphic image (a), the circuit realization (b), the output state Y = 0 (c) and Y = 1 (d)

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

Table of truth of the logical multiplier (I), conditional graphic image (a), circuit implementation (b), output state Y = 0 (in) and Y = i (r)

Fig. 3.5. The truth table of the multiplier (AND) logical element, the conditional graphic image (a), ), exit state Y = 0 (in) < strong> and 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.

Representation of basic logical operations in the form of time diagrams

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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