The implication (logical consequence)

The composite proposition of such structure names as an implication of those two partial propositions, from which one it is derived. From them first, the supplied word «if», is named as premise, or foundation, second, begin with a word that, — corollary, or conclusion.

Implication is designated as A→B.

It is read as “if A then B”, “if A else B”;”A implicated B”.

From them first, the supplied word «if», is named as premise second, begin with a word «then», — corollary.

A truth table for strict definition of the relevant logic operation also called as an implication.

Exclusive disjunction (exclusive “or”, strict disjunction, symmetric difference)

Exclusive disjunction is designated as AÅ B. It is read as “or A or B“.

Equivalention (is necessary and enough) (equivalence, identity)

A logic operation named «when and in only case when» (versions: «it is necessary and enough», «if and only if»).

The composite proposition derived from two initial with the help of this operation, is named them equivalention; as this operation is named also.

Equivalention of two propositions is true in only case when, when either both propositions are true or both are false, that is the truth values that and other coincides.

Nonconjunction (Sheffer function, stroke function)

Sheffer-stroke operation, NAND operation, NOT-AND operation. It is designated as: A | B

It is mean that result is true, if false both A and B.

Antidisjunction (Pirs’s arrow, Webb’s function, Dagger’s function )

It is designated as: A О B . It is mean that result is true, if false that A and B.

Summary

In logic formula signs of logic operations are so appreciable, as characters symbolizing partial propositions. The logic structure of the composite proposition thus is visible clearly, and the possible double meanings are eliminated by application of brackets.

The true and lie — there is all multitude, can accept which one statement variable.

If two characters, which coherent with signs of a conjunction or disjunction, implication or equivalention, be considered as statement variables, each such pair will define by itself two-place function of a propositional calculus.

Boolean Algebra

Boolean algebra is basis of logic minimization.

• identities:

X + 0 = X X • 1 = X

X + 1 = 1 X • 0 = 0

• idempotence law:

X + X = X X • X = X

• complements law:

X + X′ = 1 X • X′ = 0

• commutative law:

(Latin commutativus - varying, exposed to transition) – he law of logic theory on which, on the analogies of algebra, the outcome of the operation effected above two expressions, does not depend on order of these expressions. As in logic theory it is possible to multiply expressions (in conjunction) and to add (in the disjunction), that, according to the law of a commutability the outcome of addition (multiplying) does not depend on the order of summands (multipliers) and, therefore, operation of addition (and also multiplying), i.e. conjunction and the disjunction of expressions, is commutative.

X + Y = Y + X X • Y = Y • X

• associative law:

(X + Y) + Z= X + (Y + Z)= X + Y + Z

(X • Y) • Z = X • (Y • Z) = X • Y • Z

• distributive law:

X • (Y+Z) = (X•Y) + (X•Z)

X + (Y•Z) = (X+Y) • (X+Z)