Translating a sentence into a logical expression.

Question

I am having trouble understanding the solution given for a problem in my discrete mathematics text book. Any help would be much appreciated.

Question: Let L(x, y) be the statement "x loves y", where the universe of discourse for both x and y is the set of all people in the world. Use quantifiers to express the following statement.

There is exactly one person who everybody loves.

Solution: $\exists x(\forall yL(y,x)\wedge \forall z((\forall wL(w,z))\rightarrow z=x))$

Answer 1

Essentially, this says that there is someone x whom everyone loves, and anyone who is loved by all is x.

This can, equivalently be written as follows: $$\exists x(\forall yL(y,x)\wedge \forall z(z\neq x \rightarrow \exists w(\lnot L(w, z))$$

There is someone x who everyone loves, and everyone that is not $x$ is not liked by someone.

Answer 2

I would cut the logical expression into three parts:

• "$\exists x(\forall yL(y,x) \dots$" : There exist a person $x$ who everybody loves

• "$\wedge \forall z((\forall wL(w,z)) \dots$" : and if another person $z$ is loved by every one

• "$\rightarrow z=x))$" : then $z$ is $x$.

Answer 3

$$\exists x \forall y L(x,y) \wedge \forall x\forall z (\forall y (L(x,y) \wedge L(z,y)) \rightarrow x=z)$$