Basics of logical algebra

Аюпов Равшан Хамдамович

Plan:

1. About logic

2. Logical expressions and validity tables

3. Logical elements, predicates and Logical

functions

4. Logical laws and rules of transformation of log.

expressions

5. Logical bases of the device of the computer

6. Boolean operations on polygons

1. Thinking forms

The logic is a science about forms and ways

of thinking. Main forms of thinking: 1 – Concept

2 – Statement

3 – Conclusion

1.1. Concept The concept is the form of thinking

fixing the main, essential signs of object.

Concept

Contents Volume

Set of essential signs

of object

Set of subjects on

which the concept

extends

1.2. Statement The statement is a form of thinking in which something is

approved or denied about properties of real subjects and

the relations between them.

The statement is the narrative offer.

Statement

True False

Communication of

concepts correctly

reflects properties and

the relations of real

things

The statement doesn't

correspond to reality

Statement

Simple Compound

1.3. Conclusion

The conclusion is a form of thinking by

means of which of one or several

judgments (parcels) the new judgment

(decision) can be received.

Parcels – only true judgments.

2. Propositional algebra

The algebra of statements serves for

determination of the validity or falsehood of

compound statements.

Statements are designated by names of

logical variables which can accept only two

values: "truth" (1) and "lie" (0).

Logical operation

Logical operation — in programming operation over expressions of the logical (Boolean) type, corresponding to some operation over statements in algebra of logic. As well as statements, can accept logical expressions one or the other truth conditional values — "truly" or "in a false manner". Logical operations serve for receiving difficult logical expressions from simpler. In turn, logical expressions are usually used as a condition for management of sequence of implementation of the program.

Logical operation

Any digital computer consists of logical schemes - such schemes which can be only in one of two possible conditions - or "logical zero", or "logical unit". For logical 0 and logical 1 it is possible to accept any expression, including verbal which can be characterized as "truth" and "lie". In computer facilities logical 0 and 1 is a condition of electric circuits with certain parameters. So, for logical elements and the schemes executed on technologies of transistor-transistor logic (TTL-scheme), logical 0 is tension in the range of 0 … + 0,4 B, and logical 1 is tension in the range + 2,4 … + 5 V [1]. Work of logical schemes is described by means of special mathematical apparatus which is called as logical (Boolean) algebra or algebra of logic. The Boolean algebra was developed by George Boulle (1815 - 1864), it is a basis of all methods of simplification of Boolean expressions.

Logical operation

In some programming languages (for

example in C) instead of logical type or at

the same time with it numerical types are

used. In this case it is considered that other

than zero value corresponds to logical

truth, and zero — logical lie.

Among logical operations are most known conjunction (& &), a disjunction (||), denial (!) . Them quite often confuse to bit operations though it is different things. For example, the following code in language C:

if (action_required && some_condition())

{ /* any activities */ }

won't execute a call of the subprogramme of some_condition

() if value of the logical action_required variable is false. At

such operation the second argument of operation & also

won't be calculated at all.

Logical operation

2.1 . Logical multiplication (conjunction)

2.2 . Logical addition (disjunction)

2.3 . Logical denial (inversion)

2.1. Logical multiplication (conjunction)

Association of two (or several) statements in one by means of the and union.

The compound statement is true only when truth both simple statements.

Corresponds to the union and

Designation &, ^

In the and programming languages and;

Validity table

A B F=A&B

0 0 0

0 1 0

1 0 0

1 1 1

2.2. Logical addition (disjunction) Association of two (or several) statements in one by means of the or union.

The compound statement is true only when truly at least one or the other simple

statements.

Corresponds to the OR union

Designation of V

In the or programming languages or

Validity table

A B F=AvB

0 0 0

0 1 1

1 0 1

1 1 1

2.3. Logical denial (inversion) Particle accession "not" to the statement.

Inversion does the true statement false and, on the contrary.

Corresponds to NOT union

Designation Ā

In not programming languages not

Validity table

A F= Ā

0 1

1 0

3. Logical expressions and validity tables

Logical expression – a formula into which logical variables and signs of logical

operations enter.

Example:

)(&)( BABAF

For logical expression it is possible to construct the table of the validity which

defines its validity or falsehood at all possible combinations of reference values

of simple statements.

Logical elements Logical elements — the devices intended for information processing in a

digital form (sequence of signals high — "1" and low — "0" levels in binary logic, sequence "0", "1" and "2" in ternary logic, sequence "0", "1", "2", "3", "4", "5", "6", "7", "8" and "9" in decimal logic). Physically logical elements can be executed mechanical, electromechanical (on electromagnetic relays), electronic (on diodes and transistors), pneumatic, hydraulic, optical, etc.

With electrical equipment development from mechanical logical elements passed to electromechanical logical elements (on electromagnetic relays), and then to electronic logical elements on electron tubes, later — on transistors. After the proof in 1946 of the theorem of John von Neumann about profitability of indicative position notations it became known of advantages of binary and ternary notations in comparison with decimal notation. From decimal logical elements passed to binary logical elements. The binary and ternary allows to reduce considerably number of operations and the elements which are carrying out this processing, in comparison with decimal logical elements.

With electrical equipment development from mechanical logical elements passed to electromechanical logical elements (on electromagnetic relays), and then to electronic logical elements on electron tubes, later — on transistors. After the proof in 1946 of the theorem of John von Neumann about profitability of indicative position notations it became known of advantages of binary and ternary notations in comparison with decimal notation. From decimal logical elements passed to binary logical elements. The binary and ternary allows to reduce considerably number of operations and the elements which are carrying out this processing, in comparison with decimal logical elements.

Logical elements carry out logical function (operation) over entrance signals (operands, data).

Predicate Predicate (n-seater) — is function with a set of values (or { lie, truth }), defined on a

set. Thus, each set of elements of a set of M is characterized either as "true", or as "false".

The predicate can be connected with the mathematical relation: if (m1, m2... mn) belongs to the relation, the predicate will return on it 1. In particular, the single predicate defines the accessory relation to some set.

Predicate — one of elements of logic of the first and the highest orders. Since logic of the second order, in formulas it is possible to put quantifiers on predicates.

if on any set of arguments it accepts value 1.

Predicate call identical and false

if on any set of arguments it accepts value 0.

Predicate call feasible if at least on one set of arguments it accepts value 1.

As predicates accept only two values, all operations of Boolean algebra are applicable to them, for example: denial, implication, conjunction, disjunction

4. Logical function

Any compound statement can be

considered as the logical function F(X1, X2

function, …, Xn),

where X1, X2, …, Xn – simple statements.

Function and arguments can accept only

two various values: "truth" (1) and "lie" (0).

Tables of the validity of logical functions of two arguments

argumen

ts Logical function

A B F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16

0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1

1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

Logical following (implication)

Corresponds to a turn If …, then …

designation А→В In the if programming languages if … then …

Implication is formed by connection of two statements in one by means of a

turn of speech "if …, then …".

Implication is false only when from the true first statement (precondition)

the fallacy (the second statement) follows.

Validity table

A B F14=A→B

0 0 1

0 1 1

1 0 0

1 1 1

Logical equality (equivalence) Equivalence is formed by connection of two statements in one by means of a turn

of speech "… in only case when, when …".

The compound statement formed by means of logical operation of equivalence

truly in only case when when both statements at the same time either are false, or

are true.

Validity table

A B F10

0 0 1

0 1 0

1 0 0

1 1 1

Corresponds to a turn in only case when, when …

Designation of А≡В, А~B

5. Logical laws and rules of transformation of logical expressions

0A&A

1AA

Identity law.

Any statement is identical to itself. А=А

Not contradiction law.

The statement can't be at the same time true and false.

The law excluded the third.

The statement can be either true, or false, the third isn't given.

Law of double denial.

If twice to deny some statement, we will receive the initial statement.

AA

Logical laws and rules of transformation of logical expressions

De Morgan's laws.

BAB&A

B&ABA

Commutativity law

A & B = B & A

A v B = B v A

Associativity law

(A & B) & C = A & (B & C)

(A v B) v C = A v (B v C)

Distributivity law.

(A & B) v (A & C) = A & (B v C)

(A v B) & (A v C) = A v (B & C)

6 . Logical bases of the device of the computer

Basic logical Elements

Logical element

"and"

Logical element

"OR"

Logical element

"NOT"

and

А(0,0,1,1)

В(0,1,0,1)

F(0,0,0,1)

or А(0,0,1,1)

В(0,1,0,1)

F(0,1,1,1)

not А(0,1) F(1,0)

Слагаемые Перенос Сумма

A B P S

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

Adder of binary numbers

Half-adder.

A, B – the composed

P – transfer

S – amount

)B&A(&B)(AS

P = A & B

Logical bases of the device of the computer

Logical bases of the device of the computer

Adder of binary numbers

Half-adder.

Table of the validity of logical function )B&A(&B)(AF

A B AvB A&B

0 0 0 0 1 0

0 1 1 0 1 1

1 0 1 0 1 1

1 1 1 1 0 0

)B&A(&B)(AF )B&A(&B)(AF

and

or

not and

A

B A&B

AvB

)B&A(&B)(AF

A&B

)B&A(&B)(AF

Logical bases of the device of the computer

Adder of binary numbers

Full one-digit adder

Has three entrances: A, B – composed, P0 – transfer from the younger category;

two exits: S – the sum, P – transfer.

Addition table

The

composed

Transfer from the

younger category transfer amout

A B P0 P S

0 0 0 0 0

0 1 0 0 1

1 0 0 0 1

1 1 0 1 0

0 0 1 0 1

0 1 1 1 0

1 0 1 1 0

1 1 1 1 1

P=(A&B)v(A&P0)v(B&P0)

S=(AvBvP0)&P0v(A&B&P0)

Logical bases of the device of the computer

Trigger

The trigger allows to remember, store, read out information.

The trigger stores 1 bit of information.

or not

or not

S(1) 1

1

0

R

0 1

Q

Boolean operations on polygons

Boolean operations on polygons are a

set of Boolean operations (AND, OR, NOT,

XOR, ...) operating on one or more sets

of polygons in computer graphics. These

sets of operations are widely used

in computer graphics, CAD, and

in EDA (in integrated circuit physical design

and verification software).