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Truth Conditions for And and Or

A free Logic lesson from the “And, Or, and Basic Connectives” unit, with a worked example and practice problems including step-by-step solutions.

Truth conditions are the rules for a connective. For and, every part must work. For inclusive or, at least one part must work. Learning objective: State exactly when and/or statements are true. 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: p ∧ q is true only when p and q are both true. Example 2: p ∨ q is true when p is true, q is true, or both are 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: And/or logic appears in eligibility rules, solution-set descriptions, search filters, and coding conditions.

Worked example

Problem. Example case A (Truth Conditions for And and Or): Worked example: p ∧ q is true in exactly which case?

  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

Practice problems

1. Practice case A (Truth Conditions for And and Or): 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

2. Practice case B (Truth Conditions for And and 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 Conditions for And and 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 Conditions for And and 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 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 Conditions for And and 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 D (Truth Conditions for And and 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

5. Practice case E (Truth Conditions for And and Or): Practice: Suppose p is False and q is True. 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 = False and q = True, 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.

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