CMClearMathAcademy

Truth Tables for And

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

An and column is true only on rows where both component statements are true. Learning objective: Complete truth-table rows for conjunction. 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 And): 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 And): 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 and.
  4. An and statement is true only when both parts are true.

Answer: True

Practice problems

1. Practice case A (Truth Tables for And): 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 And): 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 and.
  4. An and statement is true only when both parts are 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 And): Practice: p ∧ q is true in exactly which case?

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

Show solution
  1. An and statement needs every part true.
  2. One false part makes the whole thing false.
  3. So p ∧ q is true only when both are true.

Answer: When both p and q are true

3. Practice case C (Truth Tables for And): 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 And): 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 and.
  4. An and statement is true only when both parts are true.
  5. So p ∧ q is False.
  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: False

4. Practice case D (Truth Tables for And): 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 And): 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 And): 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 and.
  4. An and statement is true only when both parts are true.
  5. So p ∧ q is False.
  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: False

Practice this interactively with instant feedback and an AI tutor.

Practice Truth Tables for And Take the free placement check

More Logic lessons