Proof by Cases
Split into exhaustive cases.
Key idea
Split into exhaustive cases.
In depth
Split the situation into exhaustive cases and prove the claim in each. Make sure the cases cover every possibility. Useful for absolute values and parity.
Worked example
n² + n is even: if n is even, both terms are even; if n is odd, both are odd, and their sum is even.
Real-world uses
- Checking all possible remainders when testing a formula, such as n² mod 3 in algorithms
Common mistake
Missing a case, such as zero.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Proof by Cases?
Split into exhaustive cases. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Proof by Cases used in real life?
Real-world examples: Checking all possible remainders when testing a formula, such as n² mod 3 in algorithms.
What should I learn before Proof by Cases?
Make sure you are comfortable with Proof by Contradiction, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Proof by Cases?
Missing a case, such as zero.
How can I practise Proof by Cases?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Proof by Cases?
Move on to Induction Principle next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.