CMClearMathAcademy

Disjunction: "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.

In standard mathematics, or is inclusive. A statement p or q is true when p is true, q is true, or both are true. Learning objective: Evaluate and translate statements joined by 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: 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 (Disjunction: "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 (Disjunction: "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 (Disjunction: "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 (Disjunction: "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 (Disjunction: "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 (Disjunction: "Or"): Practice: In standard mathematical logic, what does p ∨ q allow?

Choices: p true, q true, or both true · exactly one of p and q, never both · only p true · neither p nor q

Show solution
  1. ∨ is inclusive or unless a problem says otherwise.
  2. Inclusive or allows both parts to be true.
  3. Exclusive or is the one that means exactly one.

Answer: p true, q true, or both true

4. Practice case D (Disjunction: "Or"): 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 (Disjunction: "Or"): Practice: Translate p ∧ q into plain English.

Choices: p and q · p or q · not p · if p then q

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

Answer: p and q

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