Relations / Function Types
Least You Need to Know: Injective, Surjective, and Bijective Functions
A function can fail by hitting two inputs with the same output, by missing outputs in the codomain, or by doing both.
Least you need to know
- Injective means distinct inputs give distinct outputs.
- Surjective means every element of the codomain is hit.
- Bijective means both injective and surjective.
- You test injective and surjective separately.
- The codomain matters for surjectivity.
Key notation
- f:A→B — function from domain A to codomain B
- injective — one-to-one
- surjective — onto
Worked example
- Let
f:{1,2,3}→{a,b,c}withf(1)=a,f(2)=b,f(3)=c.\n- Every codomain value is hit and no two inputs share an output.\n- Sofis bijective.
Common mistakes
- Students often test surjectivity against the image instead of the codomain.
- Students confuse 'not injective' with 'not a function'.
- A finite example can be injective but fail surjectivity if the codomain is larger.
How to recognize it
- The prompt names domain and codomain explicitly.
- The question asks whether values are repeated or missed.
- Function tables and small finite maps are ideal places to test injective versus surjective.
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