Determining properties of a relation on the set of all propositions.

Question

Let $P$ denote the set of all propositions.

Define a relation $R$ on $P$ by $φRψ$ if and only if $φ ⇔ ¬ψ$

Determine the properties of the above relation;

reflexivity, Symmetry, antisymmetry, transivity

Answer 1

Let $P$ denote the set of all propositions. Define a relation $R$ on $P$ by $φRψ$ if and only if $φ⇔¬ψ$

Determining properties:

Reflexive:

$φ \not⇔ ¬ψ \Rightarrow φ\not$R$ φ $ Therefore it is not reflexive.

Symmetric:

$φRψ \rightarrow ψRφ $

Assume that $φRψ$

$φ⇔ ¬ψ$

$\Rightarrow ¬φ⇔ ¬¬ψ$

$\Rightarrow ψ⇔ ¬φ$

$\Rightarrow ψRφ$ it is therefore symmetric.

Antisymmetry:

$φ:(p\land q)$

$(p\land q)⇔¬¬(p\land q) = φRψ$

$ψ:(p\land q)$

$¬(p\land q)⇔¬(p\land q) = ψRφ$

However $(p\land q)\not⇔¬(p\land q)$ So it is not antisymmetric

Transitivity:

$φ:(p\land q)$

$ψ:¬(p\land q)$

$(φRψ)\land(ψRφ) \rightarrow φRφ? $

$φ\not$R$φ$ So it is not transitive.