In mathematics, special symbols are used to shorten the record and express the statement more accurately.
Mathematical Symbols:
For example, using the symbol " > » to numbers a, b, we get the entry " a > b”, which is an abbreviation for the sentence: “number a more number b". If - designations of lines, then the record is a statement that is parallel. Record " x M" means that x is an element of the set M.
Along with mathematical symbolism, logical symbolism is widely used in mathematics, applied to statements and predicates .
Under saying meaning a sentence that is either only true or only false. For example, the statement "–3 > 0" is false, and the statement "2 2 = 4" is true. We will designate statements in capital Latin letters, possibly with indices. For example, A= "-3 > 0», B= "2 2 = 4".
Predicate is a sentence with one variable or several variables. For example, the sentence: "number x greater than the number 0" (in characters x > 0) is a single variable predicate x, and the sentence: "a+b=c" is a three-variable predicate a, b, c.
The predicate for specific values of the variables becomes a proposition, taking on true and false values.
We will denote predicates as functions: Q(x) = « x > 0» , F(x,b,c) = « x + b = c» .
Logic symbols: .
1. Negation applies to one statement or predicate, corresponds to the particle "not" and is denoted by .
For example, the formula is an abbreviation for the sentence: "-3 is not greater than 0" ("it is not true that -3 is greater than 0").
2. Conjunction applied to two statements or predicates, corresponds to the union "and", denoted: A&B(or A B).
So the formula (–3 > 0) & (2 2 = 4) means the sentence “–3 > 0 and 2 2 = 4”, which is obviously false.
3. Disjunction applies to two statements or predicates, corresponds to the union "or" (non-separating) and is denoted A B .
Suggestion: "number x belongs to a set or a set" is represented by the formula: 
.
4. implication corresponds to the union "if ..., then ..." and is denoted: A B.
So, the entry a > –1 a > 0" is an abbreviation for the sentence "if a >-1, then a > 0».
5. Equivalence A B matches the sentence: A if and only if B».
The symbols are called quantifiers of generality and existence , respectively, apply to predicates (and not statements). The quantifier is read as "any", "every", "all", or with the preposition "for": "for any", "for all", etc. The quantifier is read: “exists”, “there is”, etc.
General quantifier applied to predicate F(x, …) containing one variable (for example, x) or several variables, resulting in the formula
1. xF(x,…), which corresponds to the sentence: "for any x performed F(x, …)» or all x have the property F(x, …)».
For example: x(x> 0) there is an abbreviation for the phrase: "any x greater than 0", which is a false statement.
2. Existence quantifier applied to the predicate F(x,…) corresponds to the sentence "there exists x, such that F(x,…)" ("there is x, for which F(x,…)") and is denoted: xF(x,…).
For example, the true statement "there is a real number whose square is 2" is written by the formula x(xR&x 2 = 2). Here the existential quantifier is applied to the predicate: F(x)= (xR&x 2 = 2) (recall that the set of all real numbers is denoted by R).
If a quantifier is applied to a predicate with one variable, then the result is a proposition, true or false. If a quantifier is applied to a predicate with two or more variables, then the result is a predicate with one less variable. So, if the predicate F(x, y) contains two variables, then in the predicate xF(x, y) one variable y(variable x is "related", you cannot substitute values for it x). To predicate xF(x, y) one can apply the quantifier of generality or existence with respect to the variable y, then the resulting formula xF(x, y) or xF(x, y) is a proposition.
So, the predicate | sin x|< a » contains two variables x, a. Predicate x(|sinx|< a) depends on one variable a, while this predicate turns into a false statement (|sinx|< ), at a= 2 we get a true statement x(|sinx|< 2).
⊃ can mean the same thing as ⇒ (the symbol can also mean a superset).
⇒ (\displaystyle\Rightarrow )
→ (\displaystyle \to )\to
⊃ (\displaystyle \supset )
⟹ (\displaystyle \implies )\implies
U+003A U+229C
:= (\displaystyle:=):=
≡ (\displaystyle \equiv )
⇔ (\displaystyle\Leftrightarrow )
The following operators are rarely supported by standard fonts. If you want to use them on your page, you should always embed the correct fonts so that the browser can display the characters without having to install fonts on your computer.
In Poland, the universal quantifier is sometimes written as ∧ (\displaystyle \wedge ), and the existence quantifier as ∨ (\displaystyle\vee ). The same is observed in German literature.
Symbolism is logical
a system of signs (symbols) used in logic to designate terms, predicates, propositions, logical functions, relations between propositions. Different logical systems can use different notation systems, so below we give only the most common symbols used in the literature on logic:
The initial letters of the Latin alphabet are usually used to denote individual constant expressions, terms;
Capital initial letters of the Latin alphabet are usually used to denote specific statements;
Letters at the end of the Latin alphabet are usually used to denote individual variables;
Uppercase letters at the end of the Latin alphabet are usually used to denote propositional variables or propositional variables; for the same purpose, small letters of the middle of the Latin alphabet are often used: p, q, r, ...;
logical symbolism; u
Signs that serve to indicate negation; read: "not", "it is not true that";
Signs for designating a conjunction - a logical connective and a statement containing such a connective as the main sign; read: "and";
A sign for designating a non-exclusive disjunction - a logical connective and a statement containing such a connective as the main sign; read: "or";
A sign to denote a strict, or exclusive, disjunction; read: "either, or";
Signs for designating an implication - a logical connective and a statement containing such a connective as the main sign; read: "if, then";
Signs to indicate the equivalence of statements; read: "if and only if";
A sign denoting the deducibility of one statement from another, from a set of statements; read: "derivable" (if the statement A is derivable from an empty set of premises, which is written as "A", then the sign " " reads: "provable");
Truth (from English true - truth); - lie (from English false - lie);
General quantifier; read "for everyone", "everyone";
Existence quantifier; read: "exists", "there is at least one";
Signs to indicate the modal operator of necessity; read: "it is necessary that";
Signs to indicate the modal possibility operator; read: "possibly".
Along with those listed in multi-valued, temporary, deontic and other systems of logic, their own specific symbols are used, however, each time it is explained what exactly this or that symbol means and how it is read (see: Logical sign).
Dictionary of logic. - M.: Tumanit, ed. center VLADOS. A.A. Ivin, A.L. Nikiforov. 1997 .
- (Logical constants) terms related to the logical form of reasoning (proof, conclusion) and are a means of conveying human thoughts and conclusions, conclusions in any field. L. to. include such words as not, and, or, there are ... Glossary of Logic Terms
GOST R ISO 22742-2006: Automatic identification. Bar coding. Linear barcode and 2D symbols on product packaging- Terminology GOST R ISO 22742 2006: Automatic identification. Bar coding. Linear barcode symbols and two-dimensional symbols on product packaging original document: 3.8 Data Matrix: Two-dimensional matrix symbology with correction ... ...
- (Wittgenstein) Ludwig (1889 1951) Austro English. philosopher, prof. philosophy at Cambridge University in 1939 1947. Philos. V.'s views were formed as under the influence of certain phenomena in the Austrian. culture early. 20th century, and as a result of creative ... ... Philosophical Encyclopedia
- (Greek logike̅́) the science of acceptable ways of reasoning. The word "L." in its modern use is ambiguous, although not as rich in semantic shades as ancient Greek. logos from which it comes. In the spirit of tradition with the concept of L ... Great Soviet Encyclopedia
- (from the Greek semeiot sign) a general theory of sign systems that studies the properties of sign complexes of a very different nature. Such systems include natural languages, written and oral, a variety of artificial languages, starting with formalized ... Philosophical Encyclopedia
This term has other meanings, see Cow (meanings). ? Domestic cow ... Wikipedia
Concept Calculus- "CALCULUS OF CONCEPTS" ("Record in concepts") the work of the German mathematician and logician Gottlob Frege, which marked the beginning of the modern form of mathematical (symbolic) logic. The full title of this work included an indication that in ... ... Encyclopedia of Epistemology and Philosophy of Science
Wittgenstein (WITTGENSTEIN) Ludwig- (1889 1951) austrian philosopher. Prof. philosophy at the University of Cambridge in 1939 47 . The philosophical views of V. were formed both under the influence of certain phenomena in the Austrian. culture of the beginning of the 20th century, and as a result of the creative development of new achievements ... ... Modern Western Philosophy. encyclopedic Dictionary
the code- 01.01.14 code [code]: A set of rules that match elements of one set with elements of another set. [ISO/IEC 2382-4, 02/04/01] Source ... Dictionary-reference book of terms of normative and technical documentation
- (Comte) founder of positivism, b. January 19, 1798 in Montpellier, where his father was a tax collector. At the Lyceum, he excelled in mathematics. Entering the Polytechnic School, he surprised professors and comrades with his mental development. AT… … Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
Conjunction or logical multiplication (in set theory, this is an intersection)
A conjunction is a complex logical expression that is true if and only if both simple expressions are true. Such a situation is possible only in a single case, in all other cases the conjunction is false.
Designation: &, $\wedge$, $\cdot$.
Truth table for conjunction
Picture 1.
Conjunction properties:
A disjunction is a complex logical expression that is almost always true, except when all expressions are false.
Designation: +, $\vee$.
Truth table for disjunction
Figure 2.
Disjunction properties:
Negation - means that the particle NOT or the word INCORRECT is added to the original logical expression, WHICH and as a result we get that if the original expression is true, then the negation of the original one will be false and vice versa, if the original expression is false, then its negation will be true.
Notation: not $A$, $\bar(A)$, $¬A$.
Truth table for inversion
Figure 3
Negative properties:
The "double negation" of $¬¬A$ is a consequence of the proposition $A$, that is, it is a tautology in formal logic and is equal to the value itself in Boolean logic.
An implication is a complex logical expression that is true in all cases except when true implies false. That is, this logical operation connects two simple logical expressions, of which the first is the condition ($A$), and the second ($A$) is the consequence of the condition ($A$).
Notation: $\to$, $\Rightarrow$.
Truth table for implication
Figure 4
Implication properties:
Equivalence is a complex logical expression that is true on equal values of variables $A$ and $B$.
Designations: $\leftrightarrow$, $\Leftrightarrow$, $\equiv$.
Truth table for equivalence
Figure 5
Equivalence properties:
A strict disjunction is true if the values of the arguments are not equal.
For electronics, this means that the implementation of circuits is possible using one typical element (although this is an expensive element).
In order to change the specified order of execution of logical operations, you must use parentheses.
For a set of $n$ booleans, there are exactly $2^n$ distinct values. The truth table for a boolean expression in $n$ variables contains $n+1$ columns and $2^n$ rows.
1.1. Notation for logical connectives (operations):
a) negation(inversion, logical NOT) is denoted by ¬ (for example, ¬A);
b) conjunction(logical multiplication, logical AND) is denoted by /\
(for example, A /\ B) or & (for example, A & B);
c) disjunction(logical addition, logical OR) is denoted by \/
(for example, A \/ B);
d) following(implication) is denoted by → (for example, A → B);
e) identity denoted by ≡ (for example, A ≡ B). The expression A ≡ B is true if and only if the values of A and B are the same (either they are both true or they are both false);
f) symbol 1 is used to denote truth (true statement); symbol 0 - to denote a lie (false statement).
1.2. Two boolean expressions containing variables are called equivalent (equivalent) if the values of these expressions are the same for any values of the variables. So, the expressions A → B and (¬A) \/ B are equivalent, but A /\ B and A \/ B are not (the meanings of the expressions are different, for example, when A \u003d 1, B \u003d 0).
1.3. Priorities of logical operations: inversion (negation), conjunction (logical multiplication), disjunction (logical addition), implication (following), identity. Thus, ¬A \/ B \/ C \/ D means the same as
((¬A) \/ B)\/ (C \/ D).
It is possible to write A \/ B \/ C instead of (A \/ B) \/ C. The same applies to the conjunction: it is possible to write A / \ B / \ C instead of (A / \ B) / \ C.
The list below is NOT meant to be exhaustive, but is hopefully representative.
2.1. General properties
2.2 Disjunction
2.3. Conjunction
2.4. Simple disjunctions and conjunctions
We call (for convenience) the conjunction simple if the subexpressions to which the conjunction is applied are distinct variables or their negations. Similarly, the disjunction is called simple if the subexpressions to which the disjunction is applied are distinct variables or their negations.
2.5. implication
