Injective, Surjective, Bijective
One-to-one, onto and both.
Key idea
One-to-one, onto and both.
In depth
Injective: different inputs give different outputs. Surjective: every element of the codomain is hit. Bijective: both, so it has an inverse and shows sets have the same size.
Worked example
f(x) = 2x on the integers is injective but not surjective. f(x) = x³ on the reals is bijective.
Real-world uses
- Checking that every password maps to a unique hash
- Matching seats to tickets exactly
Common mistake
Confusing domain, codomain and range.
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 Injective, Surjective, Bijective?
One-to-one, onto and both. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Injective, Surjective, Bijective used in real life?
Real-world examples: Checking that every password maps to a unique hash; Matching seats to tickets exactly.
What should I learn before Injective, Surjective, Bijective?
Make sure you are comfortable with Partial Orders, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Injective, Surjective, Bijective?
Confusing domain, codomain and range.
How can I practise Injective, Surjective, Bijective?
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 Injective, Surjective, Bijective?
Move on to Cardinality of Sets 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.