Problem 7 - Analysis with an introduction to proof

Question

Identify the antecedent and the consequent in each statement. (a) A sequence is convergent if it is Cauchy. (b) Convergence is a necessary condition for boundedness. (c) Orthogonality implies invertability. (d) \(K\) is closed and bounded only if \(K\) is compact.

Step-by-Step Solution

Answer

(a) Antecedent: "a sequence is Cauchy"; Consequent: "it is convergent." (b) Antecedent: "a sequence is bounded"; Consequent: "it is convergent." (c) Antecedent: "a matrix is orthogonal"; Consequent: "it is invertible." (d) Antecedent: "\(K\) is closed and bounded"; Consequent: "\(K\) is compact."

Step 1: (a) Identify the antecedent and the consequent in the statement "A sequence is convergent if it is Cauchy."

The given statement can be rewritten as "If a sequence is Cauchy, then it is convergent." Here, the antecedent is "a sequence is Cauchy," and the consequent is "it is convergent."

Step 2: (b) Identify the antecedent and the consequent in the statement "Convergence is a necessary condition for boundedness."

Rewrite the statement as "If a sequence is bounded, then it is convergent." In this statement, the antecedent is "a sequence is bounded," and the consequent is "it is convergent."

Step 3: (c) Identify the antecedent and the consequent in the statement "Orthogonality implies invertability."

Rewrite the statement as "If a matrix is orthogonal, then it is invertible." Here, the antecedent is "a matrix is orthogonal," and the consequent is "it is invertible."

Step 4: (d) Identify the antecedent and the consequent in the statement "\(K\) is closed and bounded only if \(K\) is compact."

Rewrite the statement as "If \(K\) is closed and bounded, then \(K\) is compact." In this statement, the antecedent is "\(K\) is closed and bounded," and the consequent is "\(K\) is compact."