Truth Table Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Truth Table.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A table that systematically lists every possible combination of truth values for the input variables and the resulting truth value of the expression.

List every possible combination of T/F for inputs, and compute the output.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Truth tables exhaustively define logical operators by specifying the output for every possible input combination.

Common stuck point: With n variables, you need exactly 2^n rows — forgetting any row can lead to wrong conclusions about tautologies or contradictions.

Sense of Study hint: Start by listing all input rows using a binary counting pattern (TT, TF, FT, FF). Then fill in each output column one operator at a time, left to right.

Worked Examples

Example 1

easy
Construct the truth table for p \Rightarrow q.

Solution

  1. 1
    List all combinations: (T,T), (T,F), (F,T), (F,F).
  2. 2
    Apply the rule: p \Rightarrow q is false only when p is true and q is false.
  3. 3
    Results: (T,T) \to T, (T,F) \to F, (F,T) \to T, (F,F) \to T.

Answer

\begin{array}{cc|c} p & q & p \Rightarrow q \\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \end{array}
A truth table systematically lists every possible combination of truth values for the variables and evaluates the compound statement for each.

Example 2

medium
Build a truth table to verify that p \Rightarrow q is logically equivalent to \neg p \lor q.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Construct the truth table for \neg(p \land q) and verify it equals \neg p \lor \neg q (De Morgan's Law).

Example 2

medium
Construct the truth table for p \land \neg q.
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Related Concepts

Logical Statement

Background Knowledge

These ideas may be useful before you work through the harder examples.

logical statement