Question

Assuming that Lxy is a means "x loves y" and Rx "x is rich" write logical formulas for

a) It is enough to love someone to be rich
b) being rich doesn't mean loving someone
c) you are not rich i you love no one.
d) Love is not always a symmetrical relation

Answer 1

Task. Assuming that Lxy is a means “x loves y” and Rx “x is rich” write logical formulas for

a) it is enough to love someone to be rich,

b) being rich doesn’t mean loving someone,

c) you are not rich if you love no one,

d) love is not always a symmetrical relation.

Solution.

a) The phrase

it is enough to love someone to be rich

can be reformulated as follows:

if exists y such that x loves y, then x is rich

Using logical formulas this can be written in the following way:

\forall x:\ \Big{[}\big{(}\exists y:Lxy\big{)}\Rightarrow Rx\Big{]}.

b) The phrase

being rich doesn’t mean loving someone

can not be formulated as follows:

there exists x which is rich and which loves no one.

In terms of logical formulas this means that

$\exists x:(Rx\wedge(\forall y:\overline{Lxy})).$

c) The phrase

you are not rich if you love no one

can not be formulated as follows:

if x loves no one, then x is not rich.

In terms of logical formulas this means that

$\forall x:(\forall y:\overline{Lxy})\Rightarrow\overline{Rx},$

$\forall x:\overline{(\forall y:\overline{Lxy})}\vee\overline{Rx},$

$\forall x:(\exists y:Lxy)\vee\overline{Rx}.$

d) The phrase

love is not always a symmetrical relation

can be reformulated as follows:

there exist x and y such that (x loves y) and (y does not love x).

Using logical formulas we obtain the following expression:

$\exists x\ \exists y:(Lxy\wedge\overline{Lyx}).$