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Compound Statements

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

Compound statements join simpler statements with words such as not, and, or, if, and only if. The truth of the compound statement depends on the truth of its parts and the connective being used. Learning objective: Identify statements made by combining smaller claims. 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 (Compound Statements): 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 (Compound Statements): If p is True and q is True, what is p ∧ q?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The connective ∧ means and.
  4. An and statement is true only when both parts are true.

Answer: True

Practice problems

1. Practice case A (Compound Statements): 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 (Compound Statements): If p is True and q is True, what is p ∧ q?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The connective ∧ means and.
  4. An and statement is true only when both parts are true.
  5. Here p ∧ q is True.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: True

2. Practice case B (Compound Statements): 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 B (Compound Statements): If p is True and q is True, what is p ∨ q?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The connective ∨ means inclusive or.
  4. It is true when at least one part is true.
  5. Here p ∨ q is True.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: True

3. Practice case C (Compound Statements): Which symbolic form matches "p and q"?

Choices: p ∧ q · p ∨ q · ¬p · p → q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Compound Statements): Which symbolic form matches "p and q"?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The word and is represented by ∧.
  4. Keep the statement letters in place.
  5. So p and q becomes p ∧ q.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: p ∧ q

4. Practice case D (Compound Statements): Which symbolic form matches "p or q" in standard mathematical logic?

Choices: p ∨ q · p ∧ q · ¬p · p ↔ q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Compound Statements): Which symbolic form matches "p or q" in standard mathematical logic?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The word or is represented by ∨.
  4. Standard mathematical or is inclusive.
  5. So p or q becomes p ∨ q.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: p ∨ q

5. Practice case E (Compound Statements): A club requires "age 12 or older and permission slip signed." Which connective joins the two requirements?

Choices: and · or · if and only if · not

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Compound Statements): A club requires "age 12 or older and permission slip signed." Which connective joins the two requirements?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Both requirements must be met.
  4. When two conditions are both required, use and.
  5. So the connective is and.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: and

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