CMClearMathAcademy

Compound Truth Tables

A free Logic lesson from the “Truth Tables” unit, with a worked example and practice problems including step-by-step solutions.

Compound truth tables are built from inside out. Students evaluate smaller pieces first, then use those columns to find the final column. Learning objective: Evaluate compound statements one column at a time. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: If p is false, then ¬p is true. Example 2: If p is true and q is false, then p ∧ q is false while p ∨ q is true. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: Truth tables are the bridge from language to Boolean thinking in computer science, circuits, spreadsheets, and careful casework.

Worked example

Problem. Example case A (Compound Truth Tables): A truth table with three variables has how many rows?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Compound Truth Tables): A truth table with three variables has how many rows?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Each variable has two truth values.
  4. Three variables create 2 x 2 x 2 cases.

Answer: 8

Practice problems

1. Practice case A (Compound Truth Tables): A truth table with three variables has how many rows?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Compound Truth Tables): A truth table with three variables has how many rows?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Each variable has two truth values.
  4. Three variables create 2 x 2 x 2 cases.
  5. That gives 8 rows.
  6. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

2. Practice case B (Compound Truth Tables): A truth table with two variables has how many rows?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (Compound Truth Tables): A truth table with two variables has how many rows?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Each variable has two truth values.
  4. Two variables create 2 x 2 cases.
  5. That gives 4 rows.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. Practice case C (Compound Truth Tables): In the row p=True, q=False, r=True, what is p ∨ q?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Compound Truth Tables): In the row p=True, q=False, r=True, what is p ∨ q?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. p is True and q is False.
  4. An inclusive-or statement is true when at least one part is true.
  5. The final value is True.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: True

4. Practice case D (Compound Truth Tables): In the row p=True, q=False, r=False, what is p → q?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Compound Truth Tables): In the row p=True, q=False, r=False, what is p → q?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. p is True and q is False.
  4. A conditional is false only when p is true and q is false.
  5. The final value is False.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: False

5. Practice case E (Compound Truth Tables): In the row p=False, q=True, r=True, what is p ↔ q?

Choices: True · False

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Compound Truth Tables): In the row p=False, q=True, r=True, what is p ↔ q?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. p is False and q is True.
  4. A biconditional is true when both parts have the same truth value.
  5. The final value is False.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: False

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