A student constructs a truth table for the statement P ∧ Q → R and writes only 4 rows, reasoning that she only sees two variables, P and Q. What is wrong with her table?
AShe used the wrong connective — P ∧ Q → R requires a biconditional, not a conditional
BThe table has too few rows: with three atomic variables (P, Q, and R), a complete truth table requires 2³ = 8 rows
CNothing is wrong — P ∧ Q counts as one compound variable, so 4 rows is correct
DShe should have used operator precedence to simplify the formula before building the table
P ∧ Q → R contains three distinct atomic variables: P, Q, and R. Each can be true or false independently, giving 2³ = 8 possible combinations. The student's error is treating P ∧ Q as a single unit and ignoring that R is an independent variable. A table with only 4 rows leaves 4 interpretations unchecked — it is logically incomplete, and any conclusion drawn from it would be unwarranted. The rule is strict: n atomic variables requires exactly 2ⁿ rows.
Question 2 Multiple Choice
You need to evaluate the statement P ∧ Q ∨ R (no parentheses). Which sub-expression should be computed first, and why?
AP ∧ Q, because ∧ has higher precedence than ∨ and binds its operands more tightly
BP ∧ Q ∨ R left to right, since logical formulas are read like arithmetic from left to right
CQ ∨ R, because disjunction applies to the last two variables
DThe entire expression at once — precedence rules only apply when there are parentheses
Operator precedence in logic follows a strict hierarchy: ¬ binds most tightly, then ∧, then ∨, then →, then ↔. Because ∧ has higher precedence than ∨, P ∧ Q ∨ R is parsed as (P ∧ Q) ∨ R, not P ∧ (Q ∨ R). These two formulas have different truth tables, so applying left-to-right reading (option B) instead of precedence would produce incorrect results. Adding intermediate columns for each subexpression keeps evaluation organized and correct.
Question 3 True / False
Two compound statements are logically equivalent if and only if they produce identical truth values in every row of their truth tables.
TTrue
FFalse
Answer: True
Logical equivalence (written A ≡ B) is defined exactly by this: the two statements must agree on every possible truth value assignment to the atomic variables. If even one row differs, the statements are not equivalent. This is how truth tables are used to verify equivalences like P → Q ≡ ¬P ∨ Q — you build both columns and check that they match in every row. Truth tables are definitive for this purpose on small formulas.
Question 4 True / False
When constructing a truth table, you should evaluate connectives from left to right across the formula, the same way you would read a sentence.
TTrue
FFalse
Answer: False
Left-to-right evaluation is the most common computational error in truth table construction. Connectives must be evaluated according to operator precedence: ¬ first (tightest binding), then ∧, then ∨, then →, then ↔. Parentheses override the default precedence. For example, ¬P ∨ Q ∧ R must be evaluated as ¬P ∨ (Q ∧ R), not (¬P ∨ Q) ∧ R. Building intermediate columns — one per subexpression in precedence order — is the correct method.
Question 5 Short Answer
Why must a truth table for a compound statement with n atomic variables have exactly 2ⁿ rows, and what is the consequence of constructing a table with fewer?
Think about your answer, then reveal below.
Model answer: Each atomic variable can independently be either true or false — 2 possible values. With n independent variables, the number of distinct combinations is 2 × 2 × ... × 2 (n times) = 2ⁿ. Each row represents one interpretation — one complete assignment of truth values to all variables. A table with fewer rows omits some interpretations, making it logically incomplete. Any conclusion drawn from an incomplete table — such as claiming two formulas are equivalent or that a formula is a tautology — is invalid because the unchecked rows might contain a counterexample.
The requirement for completeness is not a technicality — it is the entire point of truth tables. Their power comes from the fact that they leave no case unchecked. An incomplete table provides false confidence: it looks systematic but fails the fundamental test of exhaustiveness.