Question

Let K(x, y) denote the statement "x knows y" and D denote the domain of all people. Express the following English sentences as a quantified proposition using the definitions above:

1. Everybody knows somebody.
2. There is somebody that no one knows.
3. There is no one who knows everybody

Answer 1

Given: K(x,y) denotes statement:

"x knows y"

D denote domain of all people.

Step-by-step explanation:

1. Everybody knows somebody.

Solution:

∀x∈D ∃y∈D : K(x, y)

∀ means for all. Here it is used for Everybody.

∃ means there exists some. Here it represents Somebody.

∈ means belongs to . Both x and y belongs to the domain D of all people.

2. There is somebody that no one knows.

Solution:

∀x∈D ∃y∈D : ¬K(x, y)

∀ means for all. ∃ means there exists some. ∈ means belongs to both x and y belongs to the domain D of all people.¬ this is negation sign which means not K(x,y). So the negation of everybody knows somebody can be expressed as there is somebody that no one knows.

3. There is no one who knows everybody

Solution:

This can be represented in both the ways below.

∀ y∈D ∃ x∈D : K(x, y)

∀ means for all. ∃ means there exists some. ∈ means belongs to both x and y belongs to the domain D of all people.

∀x∈D ∀y∈D : ¬K(x, y)

∀ means for all. The negation shows that there is no one who knows everybody.