For number $1,$ use the following predicates:

The domain is the set of all vectors.

$L(x,y)$ : The vectors $x$ and $y$ are linearly independent.

$D(x,y)$ : The determinant of the matrix made up of $x$ and $y$ equals zero.

$S(x,y)$:$x$ and $y$ are scalar multiples of each other.

Translate the following statements into logic:

a$.$ The determinant for all vectors $x$ and $y$ that are not linearly independent is equal to zero.

b$.$ If $x$ and $y$ are linearly independent, then they are not scalar multiples of each other.

c$.$ There is a set of vectors $x$ and $y$ that are linearly independent and are not scalar multiples of each other.