Chapter 8.4, Problem 22E

a)

To determine

To mark:

The given statement, “If d is a real number and f is a nonzero linear functional defined on ${\text{R}}^n$, then $\left[f:d\right]$ is a hyperplane in ${\text{R}}^n$” as true or false.

b)

To determine

To mark:

The given statement, “Given any vector n and any real number d, the set $\left\{\text{x}:\text{n}\cdot \text{x}=d\right\}$ is a hyperplane” as true or false.

c)

To determine

To mark:

The given statement, “If A andB are nonempty disjoint sets such that A is compact and B is closed, then there exists a hyperplane that strictly seperates A and B” as true or false.

d)

To determine

To mark:

The given statement, “If there exists a hyperplane H such that H does not strictly separate two sets A and B, then $\left(\text{conv}A\right)\cap \left(\text{conv}B\right)\ne \varphi$” as true or false.