Question

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 \(B\).

Short answer

1. The given statement is False.
2. The given statement is False.
3. The given statement is True.
4. The given statement is False.

Step-by-step solution

Use the definition of linear functional

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.

 Use theorem 11

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.

 Use theorem 12

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.

 Use the concept of theorem 13

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.