Function
A function f: X → Y assigns exactly one element of Y to each element of X. X is the domain, Y the codomain, and f(X) the image.
Find a concept
REFERENCE SHEET
Sets & maps
Specify the objects and the maps between them before manipulating either.
Before this No prerequisites beyond basic mathematical notation.
f⁻¹(B) = {x ∈ X : f(x) ∈ B}
Definitions
Injective, surjective, bijective
Injective means f(x)=f(x′) implies x=x′. Surjective means f(X)=Y. Bijective means both.
Equivalence relation
A relation that is reflexive, symmetric, and transitive. Its equivalence classes partition the set.
Key theorems
Inverse criterion
A function has a two-sided inverse exactly when it is bijective.
Preimages preserve set operations
For any function, preimages preserve unions, intersections, and complements relative to the domain and codomain.
Notable applications
Quotient constructions
Identify objects related by an equivalence relation: congruence classes, quotient groups, and spaces of solutions.
Data transformations
Injectivity records whether a transformation loses information; surjectivity records whether every target can be reached.
Work it out
Try before revealing01 / Compute
For f: ℝ → ℝ, f(x)=x², find f⁻¹([1,4]).
Hint
Solve 1 ≤ x² ≤ 4.
Solution
The preimage is [−2,−1] ∪ [1,2].
02 / Prove
Show f⁻¹(A ∩ B)=f⁻¹(A) ∩ f⁻¹(B).
Hint
Follow one arbitrary element through both membership statements.
Solution
x belongs to the left side iff f(x) belongs to both A and B, iff x belongs to both preimages. Thus the sets have the same elements.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: