Question: In Exercises 21 and 22, mark each statement True or False. Justify each answer.

22 a. If \(d\)is a real number and \(f\) is a nonzero linear functional defined on \({\mathbb{R}^n}\) , then \(f:d\)is a hyperplane in \({\mathbb{R}^n}\) .

b. Given any vector n and any real number \(d\), the set \(\left\{ {x:n \cdot x = d} \right\}\) is a hyperplane.

c. If \(A\) and \(B\) are nonempty disjoint sets such that \(A\) is compact and \(B\) is closed, then there exists a hyperplane that strictly separates \(A\) and \(B\).

d. If there exists a hyperplane \(H\) such that \(H\) does not strictly separate two sets \(A\) and \(B\), then \(\left( {{\rm{conv}}\,A} \right) \cap \left( {{\rm{conv}}\,B} \right) \ne \emptyset \) and \(B\).

For each scalar \(d\) in \(\mathbb{R}\), the symbol \(\left[ {f:d} \right]\) denotes the set of all \(x\) in \({\mathbb{R}^n}\) at which the value of \(f\) is \(d\).

So, the statement in (a) is true.

According to theorem 11, if \(H\) is a hyperplane, there exists a nonzero vector \(n\) and a real number\(d\)such that \(H = \left\{ {x:n \cdot x = d} \right\}\).

So, the statement in (b) is false.

According to theorem 12, the sets \(A\) and \(B\) must be nonempty and convex for a hyperplane \(H\) that strictly separates \(A\) and \(B\).

So, the statement in (c) is false.

If no hyperplane \(H\) strictly separates \(A\) and \(B\), it implies that their convex hulls intersect. It might be that some other hyperplane not parallel to H would strictly separate them.

So, the statement in (d) is false.