Truth Tables

A truth table lists the value of a compound proposition for every possible combination of values of its variables. It is the most reliable way to evaluate or compare logical expressions.

How many rows?

With $n$ variables, each can be T or F, so there are $2^n$ rows. Two variables need 4 rows, three need 8.

Building a table step by step

1List all 2ⁿ combinations of the variables

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2Add a column for each sub-expression, innermost first

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3Fill each column from the columns to its left

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4The last column is the value of the whole expression

Building a truth table

Worked example: $(p \to q) \land \lnot p$

$p$	$q$	$p \to q$	$\lnot p$	$(p \to q) \land \lnot p$
T	T	T	F	F
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

Reading the last column: the expression is true exactly when $p$ is false.

Common mistakes

• Missing rows. Always check you have $2^n$ rows.
• Evaluating the outer connective before the inner ones. Work from the innermost brackets outwards.

Key takeaways

• A truth table covers every combination of truth values: $2^n$ rows for $n$ variables.
• Add one column per sub-expression and evaluate from the inside out.
• The final column gives the value of the whole proposition in each case.

Test yourself

4 questions. Pick an answer to check it, or open "Show answer".

1. How many rows does a truth table for a proposition with 3 variables have?

   • A3
   • B6
   • C8
   • D9
   Show answer

   Answer: C. 8

   Explanation: Each variable has 2 possible values, so 3 variables give $2^3 = 8$ combinations. 6 comes from wrongly multiplying $3 \times 2$.

2. In the truth table for $p \land q$, in how many rows is the result true?

   • A1
   • B2
   • C3
   • D4
   Show answer

   Answer: A. 1

   Explanation: Conjunction is true only when both $p$ and $q$ are true, which happens in exactly one of the 4 rows.

3. What is the value of $\lnot p \lor q$ when $p$ is T and $q$ is F?

   • ATrue
   • BFalse
   • CCannot be determined
   • DBoth true and false
   Show answer

   Answer: B. False

   Explanation: $\lnot p$ is F and $q$ is F, and F $\lor$ F is F. Notice this matches $p \to q$ in the same row, which is not a coincidence.

4. When building a truth table for $\lnot(p \land q)$, which column should you fill first after $p$ and $q$?

   • A$\lnot(p \land q)$
   • B$\lnot p$
   • C$p \land q$
   • D$\lnot q$
   Show answer

   Answer: C. $p \land q$

   Explanation: Work from the inside out. The negation applies to the whole conjunction, so you need $p \land q$ before you can negate it. $\lnot p$ and $\lnot q$ don’t appear in the expression at all.