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Compound Claims in Math Word Problems

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

Conjunction requires both parts; inclusive disjunction requires at least one. First evaluate each smaller claim, then combine them in the order set by parentheses. An English requirement using both must not be weakened to either.

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. Complete the final column for p ∧ q. Row order (p,q): TT, TF, FT, FF.

  1. At T, T, p ∧ q is true.
  2. At T, F, p ∧ q is false.
  3. At F, T, p ∧ q is false.
  4. At F, F, p ∧ q is false.

Answer: T, F, F, F

Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Complete the final column for p ∧ q. Row order (p,q): TT, TF, FT, FF.

Choices: F, T, T, T · F, F, F, F · T, F, F, T · T, F, F, F

Show solution
  1. At T, T, p ∧ q is true.
  2. At T, F, p ∧ q is false.
  3. At F, T, p ∧ q is false.
  4. At F, F, p ∧ q is false.

Answer: T, F, F, F

2. Complete the final column for p ∨ q. Row order (p,q): TT, TF, FT, FF.

Choices: F, T, T, F · T, T, T, T · T, T, T, F · F, F, F, T

Show solution
  1. At T, T, p ∨ q is true.
  2. At T, F, p ∨ q is true.
  3. At F, T, p ∨ q is true.
  4. At F, F, p ∨ q is false.

Answer: T, T, T, F

3. Entry to the archive requires a badge AND a reservation. Sam has a badge but no reservation. Does Sam qualify?

Choices: The rule says nothing about this case · No · Yes · Only if the badge is blue

Show solution
  1. Both requirements must be true.
  2. The missing reservation makes the conjunction false.

Answer: No

4. An alarm activates for smoke OR heat, using inclusive or. Both sensors activate. Does the alarm activate?

Choices: Yes · No, exactly one sensor must activate · No, both cancel · Cannot tell from the rule

Show solution
  1. Inclusive or permits either condition or both.

Answer: Yes

5. A code requires (p OR q) AND r. Given p=true, q=false, r=false, what is the result?

Choices: True · Equal to p · Undefined because q is false · False

Show solution
  1. p OR q is true.
  2. True AND false is false.

Answer: False

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