To solve this problem, let's translate the English statement into symbolic logic, as instructed.

Statement Analysis:

The statement is:

"If Christina Aguilera's singing soul (C) and Justin Timberlake's singing pop (J) are sufficient and necessary conditions for Kelly Clarkson's (K) singing rock, then neither Beyoncé (B) nor Shakira (S) will do rap."

This breaks down as follows:

1. "Christina Aguilera's singing soul (C) and Justin Timberlake's singing pop (J) are sufficient and necessary conditions for Kelly Clarkson's (K) singing rock" indicates a biconditional (↔). This means $(C \land J) \leftrightarrow K$.

2. "Then neither Beyoncé (B) nor Shakira (S) will do rap" translates to $\neg (B \lor S)$, meaning "not B or S".

Symbolic Translation:

Now we combine these parts:

• The entire statement can be written as:
  $[(C \land J) \leftrightarrow K] \rightarrow \neg (B \lor S)$

Matching the Answer Choices:

The correct answer should match this symbolic form. Let's go through each option:

1. Option 1: $[(C \cdot J) \supset K] \supset \sim (B \lor S)$
   This does not match because it uses implication $(\supset)$ instead of biconditional $(\leftrightarrow)$ between $C \land J$ and $K$.

2. Option 2: $[(C \cdot J) = K] \supset \sim (B \cdot S)$
   This is incorrect because it uses $(B \cdot S)$ instead of $(B \lor S)$.

3. Option 3: $[(C \cdot J) \supset K] \supset \sim (B \cdot S)$
   This is incorrect for the same reason as Option 2 and uses implication $(\supset)$ instead of a biconditional.

4. Option 4: $[(C \lor J) = K] \supset \sim (B \lor S)$
   This is incorrect because it uses disjunction $(C \lor J)$ instead of conjunction $(C \cdot J)$.

5. Option 5: $[(C \cdot J) = K] \supset \sim (B \lor S)$
   This option matches our analysis. It uses biconditional $(=)$ between $C \cdot J$ and $K$ and correctly represents "neither B nor S" as $\sim (B \lor S)$.

Conclusion:

The correct answer is: Option 5: $[(C \cdot J) = K] \supset \sim (B \lor S)$.

Would you like further details on the symbolic logic used here or additional examples?

Related Questions:

1. What is the difference between implication and biconditional in symbolic logic?
2. How do you translate "necessary and sufficient" conditions in logic?
3. What does the symbol $\lor$ represent in logical notation?
4. How would the statement change if "either Beyoncé or Shakira will do rap" was given instead?
5. How does negation affect compound statements like $B \lor S$?

Tip:

When translating English statements to symbolic logic, pay close attention to phrases like "if and only if," which indicate a biconditional (↔), and words like "or" and "and" to use the correct logical operators.