Logical conjunction

See also: Conjunction

The logical conjunction of two events, as its name indicates it, represents the fact that two events are joint (present simultaneously).
In the Logical language or mathematical and in the technical fields which employ it, the conjunction , or AND logical , is a logical operator in the Calcul of the proposals. The proposal obtained by connecting two proposals by this operator is also called their conjunction, or produces logical to them. The conjunction of two proposals P and Q is true if the two proposals are simultaneously true; if not it is false. The conjunction is written:

P ∧ Q

and is read

“ P and Q ”

For example, let us consider:

( X > 13) ∧ ( X < 27).

If X is worth 36, then X > 13 is true, but X < 27 is false, thus this proposal is false. But if X is worth 20, then the two parts of the proposal are true, thus the conjunction is also true.

The symbol “∧” is called connector of conjunction.

The Truth table of a conjunction is given by the following table

Intuitively, the logical operator works in the same way as the common word “and”. The sentence “It rains and I am inside” affirms that two things are simultaneously true: that it rains outside, and that I am inside. Logically, this assertion would be noted has and B , if has represents the assertion “it rains”, and B replaces “I am at interior”.

The conjunction that we described is a binary Operator, which means that it combines two proposals in only one. However, we can connect conjunctions, by considering for example has ∧ B ∧ C , which is by definition one or the other of the two logically equivalent proposals ( has ∧ B ) ∧ C or has ∧ ( B ∧ C ). This proposal is true when has , B , and C is simultaneously true. The sequence of the conjunctions is made possible thanks to the Associativité of the ∧. The operator is also Commutatif; has ∧ B is equivalent to B ∧ has .

Let us give some properties of the conjunction:
Are P , Q and R three proposals.

( P ∧ P ) ⇔ P idempotence of “and”
( P ∧ Q ) ⇔ ( Q ∧ P ) commutation of “and”
(( P ∧ Q ) ∧ R ) ⇔ ( P ∧ ( Q ∧ R )) associativeness of “and”
¬ ( P ∧ Q ) ⇔ ((¬ P ) ∨ (¬ Q )) the negation of a conjunction is the disjunction of the negations
¬ ( P ∨ Q ) ⇔ ((¬ P ) ∧ (¬ Q )) the negation of a disjunction is the conjunction of the negations
( P ∨ ( Q ∧ R )) ⇔ (( P ∨ Q ) ∧ ( P ∨ R )) distributivity of “or” compared to “and”
( P ∧ ( Q ∨ R )) ⇔ (( P ∧ Q ) ∨ ( P ∧ R )) distributivity of “and” compared to “or”
¬ ( P ∧ (¬ P )) law of noncontradiction

The generalization of the conjunction to families (possibly infinite) of proposals is the universal quantification, which belongs to the Calcul of the predicates.

See too

Internal bonds

logical Disjunction

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