Combinational logic functions and Boolean algebra, logical tables, Karnaugh maps, minimization, solid execution logic and combinational logic functions
Propositional algebra

## Verdict

The term verdict is possible to imagine a statement of various people, various weather lore, definitions, declarations, etc. In mathematical logic, the term is restricted definition: "The award of each claim, which can be unambiguously assigned logical value at all times." This is specified, what is the verdict expected. Statement may be a proposition whose truth will be decided in the future.

Statement can have two values: true, false. To mark truthful statements can be used symbols: yes, true, 1, +, high; H. The false statements are used designation: no, false, 0, -, low; L.

For better understanding, are a few examples?

- "The voltage drop across the resistor is proportional to the electric current flowing through." This is a statement that is true, the value is true.
- "The inductance is equal to the product of current and voltage." This is a statement which is false, the value is false.
- "In the constellation of Orion are living sentient beings." This is a statement, and its value will be known in the future. The result will answer yes or no.
- "What's out there?" This is not a statement. Generally prefixed questions "how much" into group saying that no statement.
- "Go wash your hands!" This is not a statement, because it is an imperative sentence.

## Propositional calculus

Number of propositional understand that part of mathematical logic that examines relationships between statements only with respect to their truth and falsity. Propositional calculus does not deal with the internal structure of atomic propositions and laws according to which forms.

Character or verbal expression with which to form new statements is called functors <predicament able> [logical connectives]. Propositional statement, atomic number is called a statement without functors.

The most important functors are:

- Functors negation - a sign above the bar statement and phrase is: "no, not, not true that ...".
- Functors conjunctions – sign is A˄B a logical product of the phrase is: "and, at the same time, and".
- Functors disjunction - sign is A˅B. The phrase logical sum is: ", or, or."
- Functors implications - sign is A→B. A logical implication is: "from ... that ..., if ..., then ...!.
- Equivalence functors - sign is A ↔ B. The phrase logical equivalence is "just as, if and only if ..."

Combinations of atomic propositions are formed propositional operations from the simplest to the complex. Their significance is that they can be easily interpreted technically.

## Logic function

In classical algebra is known definition of a function: function display when one or more independent variables corresponding to one or more dependent variables. Thus, in mathematical logic to define the term logical function. Logic function is a relation between the dependent and independent variables logic. Logical variables are binary variables that take values 0 or 1

Function logic variables may be a function of one or more variables.

y = f (x1; x2; x3; ...; xn)

Each logical function can be expressed in three ways with equivalent results:

- Combining negation functors, disjunction, conjunction
- Pierce using multiple functions. Pierce function is called functor NOR. It is the negation of the disjunction of two statements.
- Sheffers using multiple functions. Sheffer function is called functor NAND. It is the negation of the conjunction of two statements. Its use is preferred for the reason that every operation in the logic function to be implemented by one technical element. Expression logical functions Sheffer function is advantageous, because it is at least technical elements.

Logic functions can solve logical table. This is a short statement of all combinations of independent variables, which may occur. Total number of options that can occur is calculated by the formula:

k = 2n

where k is the number of all possible n is the number of variables.

Kind of solution is illustrated in the following example.

### Example 4.1

Using a truth table for a logic function Y, which is a function of three variables and logic is expressed by the formula:

According to the formula is calculated as 8 choices for the independent variables.

A | B | C | Not A | Not B | A+not B | B+C | C+not A | Y
0 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0
0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1
0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 0
0 | 1 | 1 | 1 | 0 | 0 | 1 | 1 | 0
1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0
1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1
1 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0
1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1

The first three columns are the independent variables (atomic statements) which is recommended to fill gradually from the last column (in this case C) alternating values 0 and 1 The next column (in this case B) alternating values 0 and 1 at double the number of lines before the last column (in this case C). The last column (in this case A), then alternating values 0 and 1 at double the number of lines than the previous column (in this case B). In this way it is possible to proceed in the case of multiple variables. Other columns are gradually solving logic functions. The last column is the result of a logical function. To search for a combination of independent variables, it is possible to find the appropriate row resulting value logic functions.