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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): Worked example: Which rule or habit best matches Compound Statements?

  1. Compound Statements targets a specific reasoning habit.
  2. Compound Statements focuses on identify statements made by combining smaller claims.
  3. The other choices either overclaim or change the logical relationship.

Answer: Compound Statements focuses on identify statements made by combining smaller claims.

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. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 preserves the stated logical meaning and that the other choices change the rule or claim.

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. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  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 preserves the stated logical meaning and that the other choices change the rule or claim.

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 choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  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 preserves the stated logical meaning and that the other choices change the rule or claim.

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 choice with the stated logical rule, and eliminate choices that change the claim's meaning.
  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 preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: p ∨ q

5. Practice case E (Compound Statements): Lesson focus: Which transfer task would show readiness for Compound Statements?

Choices: Apply identify statements made by combining smaller claims in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.

Show solution
  1. A transfer task keeps the same objective but changes the surface context.
  2. Apply identify statements made by combining smaller claims in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply identify statements made by combining smaller claims in a new sentence or context.

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