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
- State exactly when and/or statements are true
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Truth Conditions for And and Or): Worked example: p ∧ q is true in exactly which case?
- An and statement needs every part true.
- One false part makes the whole thing false.
- So p ∧ q is true only when both are true.
Answer: When both p and q are true
Lesson preview: this page shows 5 practice items. The interactive activity contains 20 practice items and 10 quiz items (multiple choice). Counts include repeated prompts and variants.
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
- An and statement needs every part true.
- One false part makes the whole thing false.
- 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
- Inclusive or is true when at least one part is true.
- The only way it fails is if every part is false.
- 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
- ∧ means and.
- An and statement is true only when both parts are true.
- So p ∧ q is False.
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
- ∨ means inclusive or.
- It is true when at least one part is true.
- So p ∨ q is True.
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
- Evaluate ∧ (needs both) and ∨ (needs at least one) separately.
- With p = False and q = True, p ∧ q is False and p ∨ q is True.
- 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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