CMClearMathAcademy

Truth Tables for Or

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

An inclusive-or column is false only when both component statements are false. Learning objective: Complete truth-table rows for inclusive or. 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 (Truth Tables for Or): Worked example: If p is True and q is True, what is p ∨ q?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Truth Tables for Or): Worked example: If p is True and q is True, what is p ∨ q?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ∨ means inclusive or.
  4. It is true when at least one part is true.

Answer: True

Practice problems

1. Practice case A (Truth Tables for Or): Practice: If p is True and q is True, what is p ∨ q?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Truth Tables for Or): Practice: If p is True and q is True, what is p ∨ q?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ∨ means inclusive or.
  4. It is true when at least one part is true.
  5. So p ∨ q is True.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: True

2. Practice case B (Truth Tables for Or): Practice: p ∨ q (inclusive or) is false in exactly which case?

Choices: When both p and q are false · When both p and q are true · When at least one part is true · When exactly one part is true

Show solution
  1. Inclusive or is true when at least one part is true.
  2. The only way it fails is if every part is false.
  3. So p ∨ q is false only when both are false.

Answer: When both p and q are false

3. Practice case C (Truth Tables for Or): Practice: If p is True and q is False, what is p ∨ q?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Truth Tables for Or): Practice: If p is True and q is False, what is p ∨ q?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ∨ means inclusive or.
  4. It is true when at least one part is true.
  5. So p ∨ q is True.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: True

4. Practice case D (Truth Tables for Or): Practice: Suppose p is True and q is False. Comparing p ∧ q with p ∨ q, which is correct?

Choices: p ∨ q is true, but p ∧ q is false. · p ∧ q is true, but p ∨ q is false. · Both p ∧ q and p ∨ q are true. · Neither p ∧ q nor p ∨ q is true.

Show solution
  1. Evaluate ∧ (needs both) and ∨ (needs at least one) separately.
  2. With p = True and q = False, p ∧ q is False and p ∨ q is True.
  3. p ∧ q can never be true while p ∨ q is false.

Answer: p ∨ q is true, but p ∧ q is false.

5. Practice case E (Truth Tables for Or): Practice: If p is False and q is True, what is p ∨ q?

Choices: True · False

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Truth Tables for Or): Practice: If p is False and q is True, what is p ∨ q?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ∨ means inclusive or.
  4. It is true when at least one part is true.
  5. So p ∨ q is True.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: True

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