[Math] Translating a sentence into a logical expression.

discrete mathematicslogicquantifiers

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

Best Answer

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 $x$.

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