CMClearMathAcademy

Truth Tables for Not

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

Negation flips a truth value. If p is true, ¬p is false; if p is false, ¬p is true. Learning objective: Complete truth-table rows for negation. 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 Not): Worked example: If p is True, what is ¬p?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Truth Tables for Not): Worked example: If p is True, what is ¬p?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ¬ means not.
  4. Negation flips the truth value.

Answer: False

Practice problems

1. Practice case A (Truth Tables for Not): Practice: If p is True, what is ¬p?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Truth Tables for Not): Practice: If p is True, what is ¬p?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ¬ means not.
  4. Negation flips the truth value.
  5. So ¬p 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

2. Practice case B (Truth Tables for Not): Practice: If p is True, what is ¬p?

Choices: True · False

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (Truth Tables for Not): Practice: If p is True, what is ¬p?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ¬ means not.
  4. Negation flips the truth value.
  5. So ¬p 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

3. Practice case C (Truth Tables for Not): Practice: Translate ¬p into plain English.

Choices: not p · p · p and q · p or q

Show solution
  1. Read each symbol: ∧ = and, ∨ = or (inclusive), ¬ = not.
  2. ¬p reads as "not p".
  3. Keep the inclusive meaning of or unless told otherwise.

Answer: not p

4. Practice case D (Truth Tables for Not): Practice: Translate "p and not q" into symbols.

Choices: p ∧ ¬q · p ∨ ¬q · ¬p ∧ q · p ∧ q

Show solution
  1. Match each English word to its symbol: and = ∧, or = ∨, not = ¬.
  2. "p and not q" becomes p ∧ ¬q.
  3. Check that the connective and any negation are placed correctly.

Answer: p ∧ ¬q

5. Practice case E (Truth Tables for Not): Practice: If p is False, what is ¬p?

Choices: True · False

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Truth Tables for Not): Practice: If p is False, what is ¬p?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. ¬ means not.
  4. Negation flips the truth value.
  5. So ¬p 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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