Question

Use quantifiers to express the Statement below.

• Let P(x,y) be the statement. " x loves y" where D for both x and y is the set of all people in the world

• Denote the propositions given in the next slide

- Everyone loves everyone.

- For everyone, everyone loves them.

- Everyone loves someone.

- There is someone who loved by everyone.

- There is someone who loves everyone.

- For everyone, there is someone who loves them.

- There is someone who loves someone

- There is someone who is loved by someone.

Answer 1

$\forall x\forall y\,\,P\left( x,y \right) \\\forall y\forall x\,\,P\left( x,y \right) \\\forall x\exists y\,\,P\left( x,y \right) \\\exists y\forall xP\left( x,y \right) \\\exists x\forall yP\left( x,y \right) \\\forall y\exists xP\left( x,y \right) \\\exists x\exists yP\left( x,y \right) \\\exists y\exists xP\left( x,y \right) \\$