Consider the following truth table:

Which one of the following wffs correctly represents
𝑊
?

Question

Consider the following truth table:

Which one of the following wffs correctly represents 
𝑊
?

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and listing the given option clearly.
Question: Consider the truth table provided. Which of the following well-formed formulas (wffs) correctly represents W?
Answer option provided: (
Q ∧ ~P
) ∨ (
~Q ∧ ~P
)

Now, analyze the given option step by step to see what it expresses.
- Break down the expression: (Q ∧ ~P) ∨ (~Q ∧ ~P).
  This is a disjunction (OR) of two conjunctions. Each conjunction contains ~P, so ~P is a common factor.
- Factor out ~P conceptually: (~P) ∧ (Q ∨ ~Q).
  The inside (Q ∨ ~Q) is a tautology (it is always true regardless of Q).
- Therefore, the whole expression simplifies to ~P ∧ true, which is simply ~P.
- Consequently, the given option is logically equivalent to ¬P (i.e., not P), since the Q components cancel out when ORing the two branches.

Next, consider how this relates to W as described by the truth table (without seeing the table, we can only rely on the logical form):
- If the truth table for W matches ¬P (i.e., W is true exactly when P is false, independent of Q), then this option correctly represents W.
- If the truth table for W has dependence on Q (i.e., W is true for some assignments of Q when P is fixed, or false for some Q values when P is fixed), then this option would not match W.

Evaluating potential misconceptions:
- Misconception: (Q ∧ ~P) ∨ (~Q ∧ ~P) would depend on both Q and P in a more complex way. Correction: due to distributive factoring, the Q terms collapse and the expression becomes equivalent to ~P, so there is no remaining Q-dependence.
- Misconception: If one thought the expression simplified to Q ∨ ~Q (which is always true) times ~P, you might conclude the formula is always true when ~P is true, which is the same as ~P—so the same final result, but you need to show the factoring step to justify it.
- Misconception: Assuming the option represents W without comparing to the given truth table; you should explicitly verify by evaluating a few rows of P and Q to see whether the formula yields the same truth values as W.

Concrete row-check reasoning (illustrative):
- If P = T, ~P = F. Then (Q ∧ ~P) ∨ (~Q ∧ ~P) becomes (Q ∧ F) ∨ (~Q ∧ F) which is F ∨ F = F.
- If P = F, ~P = T. Then (Q ∧ ~P) ∨ (~Q ∧ ~P) becomes (Q ∧ T) ∨ (~Q ∧ T) which is Q ∨ ~Q, which is T.
- Thus the expression yields F when P is true, and T when P is false, independent of Q, i.e., it behaves exactly as ¬P.

In summary, this option represents the formula that is logically equivalent to ¬P. Whether it is a correct representation of W hinges on whether the truth table for W matches ¬P. If W is defined by the same truth pattern (true exactly when P is false), then this option is a correct representation. If not, it would differ from W for some assignments of P and Q.

MATH3066 (ND) 2025 Mid-Semester Test Practice Set C

Consider the following truth table: Which one of the following wffs correctly represents 𝑊 ?

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