Express the following statements using quantifiers

Everyone has at least one friend.

Some friendships are mutual.

Every friend of Jenny is also a friend of Ken.

Options:$($ in order of a$,$ b$,$ c$)$

Let

P

$($

x

$,$

y

$)$

be $$

x

has

y

as a friend" $($and assume friendship can be non$-$mutual:

x

can be friends with

y

without

y

being friends with

x

$)$ and let the domain be everyone in the world.

Express the following statements using quantifiers

Everyone has at least one friend.

Some friendships are mutual.

Every friend of Jenny is also a friend of Ken.

Options:$($ in order of a$,$ b$,$ c$)$

$$

x

$$

$!$

y

P

$($

x

$,$

y

$)$

$,$

$$

x

$1$

$$

y

$1$

$($

P

$($

x

$1$

$,$

y

$1$

$)$

$$

P

$($

y

$1$

$,$

x

$1$

$)$

$,$

$$

y

$($

P

$($

J

e

n

n

y

$,$

y

$)$

$$

P

$($

K

e

n

$,$

y

$)$

$)$

$.$

$$

y

$$

x

P

$($

x

$,$

y

$)$

$,$

$$

x

$1$

$$

y

$1$

$($

P

$($

x

$1$

$,$

y

$1$

$)$

$$

P

$($

y

$1$

$,$

x

$1$

$)$

$)$

$,$

$$

y

$($

P

$($

J

e

n

n

y

$,$

y

$)$

$->$

P

$($

K

e

n

$,$

y

$)$

$)$

$.$

$$

x

$$

$!$

y

P

$($

x

$,$

y

$)$

$,$

$$

x

$$

y

$($

P

$($

x

$,$

y

$)$

$->$

P

$($

y

$,$

x

$)$

$)$

$,$

$$

y

$($

P

$($

J

e

n

n

y

$,$

y

$)$

$->$

P

$($

K

e

n

$,$

y

$)$

$)$

$$

x

$$

y

P

$($

x

$,$

y

$)$

$,$

$$

x

$1$

$$

y

$1$

$($

P

$($

x

$1$

$,$

y

$1$

$)$

$$

P

$($

y

$1$

$,$

x

$1$

$)$

$)$

$,$

$$

y

$($

P

$($

J

e

n

n

y

$,$

y

$)$

$->$

P

$($

K

e

n

$,$

y

$)$

$)$