Logical equivalence

In Logique, two proposals P and Q is logically equivalent or equivalent if P and Q has simultaneously even value of truth; i.e. P and Q is true (resp. false), in exactly the same situations. One writes

P \ Leftrightarrow Q
Who is read:
“ P is true if and only if Q is true”
“ \ Leftrightarrow” is the connector of equivalence whose Truth table is given below:

Equivalence P ⇔ Q is not other than (P ⇒ Q) ∧ (Q ⇒ P) ((P implies Q) and (Q implies P)).

In other words, two proposals P and Q are equivalent amounts saying that each one of them implies the other.

In this case, the proposals “P ⇒ Q” and “Q ⇒ P” are known as Réciproque S one of the other.

To show, an equivalence P ⇔ Q, it is thus necessary to show the implication P ⇒ Q and its reciprocal.

In the natural Language, to translate that two proposals P and Q are equivalent, one will say indifferently:

  • P is true if and only if Q is true.

  • So that P is true, it is necessary and it is enough that Q is true.
  • a requirement and sufficient so that P is true is that Q is true (or Cns).
  • the truth of P is a requirement and sufficient so that Q is true.
  • P is equivalent to Q .

Other expressions “or”, “or” (but not the logical connector or), “either” can translate an equivalence as in the following example:

For any reality X , X 2= X is equivalent to X 2- X =0 is X ( X -1) =0 or (( X =0) or ( X =1))
Here, “is” (XOR) is not used to define an object, and the last “or” is one or logical (GOLD).

if ( iff ) is an abbreviation of “ if and only if ” usually used to write equivalences.

Properties

  • P ⇔ P (equivalence is reflexive)

  • (P ⇔ Q) ⇒ (Q ⇔ P) (equivalence is symmetrical)
  • (P ⇔ Q) ∧ (Q ⇔ R) ⇒ (P ⇔ R) (equivalence is transitive)
These three laws show that logical equivalence is a Relation of equivalence
  • ¬¬P ⇔ P (In the traditional Logique, this is equivalent to the principle of the excluded third)
  • (P ⇔ Q) ⇔ (¬P ⇔ ¬Q) (Contraposition)

Examples

  • There is
\ forall N \ in \ mathbb NR, N \ geq 2, \ forall X \ in \ mathbb R - \ {1 \}, (x+1) ^n= (x-1) ^n \ Leftrightarrow \ frac {(x+1) ^n} {(x-1) ^n} =1
  • equivalence ∀x, y∈ℝ (x=y ⇔ x2=y2) (while raising squared) is false because for example 22= (- 2) 2 does not imply 2=-2
  • following equivalence is true
\ forall X \ in + \ infty[, x-1 \ geq \ sqrt {x+1} \ Leftrightarrow ((x-1) ^2 \ geq x+1 \ quad \ wedge \ quad x-1 \ geq 0) (while raising squared) While raising squared, one loses information that x-1 is higher than a square Racine and must be positive and to have equivalence, one adds the property x-1>=0.

Note:

To show by equivalence is not always simple; in certain cases, it is preferable to show the reciprocal implications separately.

To say that equivalence P ⇔ Q is true does not want to say that P and Q is true, but that if one of them is true (resp. distort), the other too.

Equivalence between several proposals

That is to say three proposals P , Q and R .

To show equivalences P ⇔ Q ⇔ R, it is enough to show the implications:

P ⇒ Q, Q ⇒ R and R ⇒ P.

That is to say the Implication S P ⇒ Q, Q ⇒ R and R ⇒ P established.

To show that Q ⇒ P, one uses Q ⇒ R and R ⇒ P.

To show that R ⇒ Q, one uses R ⇒ P and P ⇒ Q.

And finally to show that P ⇒ R, one uses P ⇒ Q and Q ⇒ R.

This type of Démonstration is called a “circular” Démonstration or “in circle”.

One can generalize with N proposals P1, P2… Pn.

To show equivalences P1 ⇔ P2 ⇔… ⇔ Pn, it is enough to show the implications:

P1 ⇒ P2, P2 ⇒ P3… Pn-1 ⇒ Pn and Pn ⇒ P1.

Simple: Yew and only yew

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