Given Information
We are given with some statements, we have to prove whether they are true or False

Step-1: (a)
Statement: The columns of a matrix A are linearly independent if the equation $Ax=0$ has the trivial solution.

By Linear Independence theorem, The columns of a matrix A are linearly independent if and only if the equation $Ax=0$ has only the trivial solution. In the given statement. the word "only" is missing, so

The statement is False

Step-2: (b)
Statement: If $S$ is a linearly dependent set, then each vector is a linear combination of the other vectors in $S$.

By the definition of linearly independent vectors, a set of vectors is said to be linearly dependent, if and only if there is at least one vector in the set such that it can be written as a linear combination of the others.

However, in the statement, it is mentioned that each vector can be written as a linear combination of the other vectors, hence

The Statement if False

Step-3: (c)
Statement: The columns of any $4 \times 5$ matrix are linearly dependent.

Here the number of columns are greater than number of rows, hence there are 5 variables and 4 equations. Therefore the columns of any $4 \times 5$ matrix are linearly dependent

The Statement if True

Step-4: (d)
Statement: If $x$ and $y$ are linearly independent, and if {x,y,z} is linearly dependent, then $z$ is in Span {x,y}

If the set {x,y,z} is linearly dependent, then there should be at least one vector in the set such that it can be written as a linear combination of the other.

Hence either of the statements is True.

(1) z is in Span {x,y}
(2) y is in Span {x,z}
(3) x is in Span {y,z}
We are also given that the vectors x and y are linearly independent, so statement (1) must be true, i.e. z is in Span {x,y}

The Statement if True