Logic (elementary mathematics)

The logical is the base of the demonstrations in mathematics. Although it appears in a way hidden in all the mathematical argumentation, it is essential to the comprehension of Théorème. It is formalized in the form of Truth table in the Boolean algebra

Experimental aspect

Implications and equivalences

In logic, one discovers sufficient requirements, conditions, contraposées implications, equivalences, the reciprocal ones and. Each one of these words corresponds to a logical bond between proposals.

In the assertion:

if ABCD is a Carré then ABCD is a Parallélogramme.

It is said that

• ABCD is a square is a sufficient Condition so that ABCD is a parallelogram
• ABCD is a parallelogram is a Requirement so that ABCD is a square

One will write ABCD is a square $\backslash \; Rightarrow$ ABCD is a parallelogram. This relation is called a Implication.

One can notice that

• ABCD is a parallelogram is not a sufficient condition so that ABCD is a square
• ABCD is a square is not a requirement so that ABCD is a parallelogram

It is thus noticed that the implication

ABCD is a parallelogram $\backslash \; Rightarrow$ ABCD is a square

is a false implication.

To reverse the direction of an implication, it is to take the Réciproque.

In the case of our example, One can say that the implication

" if ABCD is a square then ABCD is a parallélogramme"

is an implication right but that its reciprocal is false.

It is noticed that in an implication, there is often loss of information between the first term of the implication and the second.

According to our preceding assertion, one can notice that

if ABCD is not a parallelogram then ABCD is not a square

To reverse the direction of the implication and to take the negation of the assertions are invited to take the Contraposée implication. In all the cases, if an implication its is just then contraposée is right.

When an implication is true and that its reciprocal is true also, it is said that the two assertions are equivalent S:

For example, for three points distinct ABC, there a:

• if ABC is a right-angled triangle is of it then AB ² + AC ² = BC ² (first implication)
• if AB ² + AC ² = BC ² then the triangle is right-angled has (reciprocal)

of it The implication and its reciprocal are true, the two assertions are thus equivalent. One will thus write

ABC is right-angled has of it if and only if (shortened in if) AB ² + AC ² = BC ²

that one shortens sometimes in

ABC is right-angled has $\backslash \; Leftrightarrow$ AB ² of it + AC ² = BC ²

Quantifiers

Actually, an assertion is not true that in a particular field, it is thus necessary to specify the field of validity and to specify for which elements it is true.

The assertion

X ² $\backslash \; geq$ 0

no direction except context has. On the other hand, one can find

• for all Real number X, X ² $\backslash \; geq$ 0 (which is a true assertion)
• for all Complexe X, X ² $\backslash \; geq$ 0 (which is a false assertion)
• It exists complexes X such as X ² $\backslash \; geq$ 0 (which is a true assertion)
• It exists the imaginary pure ones such as X ² $\backslash \; geq$ 0 (which is a false assertion)

The precise details " it existe" and " for tout" " are called Quantificateur S. Souvent implied in the reasoning, they are always essential. One can notice that for the Théorème of Pythagore quoted previously, it was specified, by a quantifier, the field of validity " for all points has, B, C distincts".

Opposite

To take the opposite of an assertion, it is to express the proposal which will be true If and only if the preceding one is false. Opposite of the sentence

" in this room, all the people are filles"

is

" in this room, there exists at least a garçon".

When one controls well logic and the quantifiers, one can take the negation of any proposal: it is enough to reverse the quantifiers and to take the contrary property.

Opposite of

" for any reality X, F (X) = F (- X) " (which translates the parity of a function definite on R )

is

" there exists a reality X such as F (X) $\backslash \; ne$ F (- X) (which translates the fact that a function defined on R is not even)

One realizes thus that there exist rules of calculation in logic which one can formalize using truth tables

Approaches a formalization

Bases

That is to say P a proposition.
It is said that P is true or false.

That is to say P a property in a state. Then not P is in an opposite state.

One can thus establish a Truth table:

And this shows that P = not P.

Now, and it is what there is of more interesting, we concentrate on the relation S:

Let us take the two basic relations: “and” and “or” (it is necessary that P is true and that Q is true, or respectively, that P either truth or Q or truth, so that the relation is true). It is said that R=P.Q= (P and Q) is true if both are true at the same time. It is said that R=P+Q= (P or Q) is true if one or the other is true.

One draws up the truth table then:

That is to say P and Q two proposals. That is to say R the relation => (it is enough that)

This is the definition of the relation implication.

In the same way, for the relation (it is necessary that)

Lastly, so that R: <=> is true (equivalence), it is necessary that => and <= are true:

The analysis of such tables enables us to show that, for example, in a demonstration, so that P<=>Q, it is necessary and it is enough that P=>Q AND not P => not Q.

Indeed:

One there has just shown that the reciprocal one of a theorem could be shown on the basis of the reverse of the assumptions to arrive contrary to the conclusion.

In the same way, it is easy to show with these truth tables that P=>Q is equivalent to nonP or Q. One will let the reader make the table in order to be convinced some.

Logic is thus at the base of mathematics, and allows them to make all the demonstrations necessary for the simplest theorems like most complex.

Important results

The following lines are “true”. They are true relations whatever P and Q two proposals.

• $non\backslash ;(not\; \backslash ,\; P)\; \backslash \; Leftrightarrow\; P$

• $(P\; \backslash \; Rightarrow\; Q)\; \backslash \; Leftrightarrow\; (not\; \backslash ,\; Q\; \backslash \; Rightarrow\; not\; \backslash ,\; P)$, the second member of equivalence is often called the contraposée of the first member.
• $(P\; \backslash \; Rightarrow\; Q)\; \backslash \; Leftrightarrow\; (Q\; \backslash ,\; or\; \backslash ,\; (not\; \backslash ,\; P))$
• $\backslash ,\; not\; \backslash ,\; (P\; \backslash \; Rightarrow\; Q)\; \backslash \; Leftrightarrow\; (P\; \backslash ,\; and\; \backslash ,\; (not\; \backslash ,\; Q))$, this result is important when one uses a reasoning by the absurdity: if one wants to prove an implication, one shows that his negation led to a nonsense.

See too

• Logical
• Axiom (elementary mathematics)

Random links:

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