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

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.
