Hypotheses and Conclusions
A free Logic lesson from the “Conditionals” unit, with a worked example and practice problems including step-by-step solutions.
An implication p→q rules out p true with q false. Its converse q→p need not follow. Its contrapositive ¬q→¬p is equivalent. In a concrete application identify which condition is actually given before applying a theorem.
What you'll learn
- Identify the if-part and then-part of a conditional
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Complete the final column for p → q. Row order (p,q): TT, TF, FT, FF.
- At T, T, p → q is true.
- At T, F, p → q is false.
- At F, T, p → q is true.
- At F, F, p → q is true.
Answer: T, F, T, T
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: T, F, T, T · F, T, F, F · F, F, T, T · T, F, T, F
Show solution
- At T, T, p → q is true.
- At T, F, p → q is false.
- At F, T, p → q is true.
- At F, F, p → q is true.
Answer: T, F, T, T
2. If an integer is divisible by 20, it is even. Which fact alone is sufficient to use this implication?
Choices: The integer is even · The integer is not divisible by the given number · The integer is positive · The integer is divisible by 20
Show solution
- The antecedent triggers the implication.
- The consequent alone does not establish the antecedent.
Answer: The integer is divisible by 20
3. The rule is “If a key is valid, the door opens.” Which observation disproves the rule?
Choices: An invalid key is used and the door stays shut · A valid key is used and the door opens · A valid key is used and the door does not open · An invalid key is used and the door opens
Show solution
- An implication fails exactly when its hypothesis is true and its conclusion false.
Answer: A valid key is used and the door does not open
4. For p→q, which statement is its converse?
Choices: p ∧ q · q → p · ¬p → ¬q · ¬q → ¬p
Show solution
- The converse swaps hypothesis and conclusion without negating them.
Answer: q → p
5. For p→q, which statement is its contrapositive?
Choices: ¬q → ¬p · q → p · ¬p → ¬q · p → ¬q
Show solution
- Swap the parts and negate both.
- This is equivalent to the original implication.
Answer: ¬q → ¬p
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