Functions · step 12 of 13

Identity, constant & inclusion functions

Three simple functions appear constantly. The identity id_A(a) = a is the unique bijection that changes nothing and acts as the neutral element for composition. A constant function sends everything to one value, so it is injective only on a one-element domain. The inclusion ι : X ↪ A of a subset is the identity rule with a bigger codomain, and it is injective but surjective only when X = A.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Identity — neutral for compositionf∘id = f and id∘f = f. It is bijective and equal to its own inverse.
Constant — collapse everythingc(a) = b₀ for all a. Its image is the single point {b₀}, so it is surjective only onto a one-element codomain.
Inclusion — same rule, bigger codomainι(x) = x for x ∈ X ⊆ A. Injective always; surjective iff X = A.
Use them to build counterexamplesThese three settle most "does property P transfer?" questions with minimal computation.

Worked example

For X = {1,2} ⊆ A = {1,2,3}, classify the inclusion ι : X → A and the constant map c : A → A with c(a) = 1.

ι(1) = 1 and ι(2) = 2: distinct inputs give distinct outputs, so ι is injective.
The image of ι is {1,2} ≠ A, so ι is not surjective.
c sends 1, 2 and 3 all to 1, so c(1) = c(2) with 1 ≠ 2 — not injective.
The image of c is {1}, so c is not surjective either.

Answer. ι is injective but not surjective; c is neither. Composing gives c∘ι, the constant map on X.

Where marks get dropped

These are the specific errors that cost credit on identity, constant & inclusion functions questions — QED's rubric penalises each of them separately.

Calling an inclusion the identity. They share a rule but differ in codomain, and that difference is exactly why one is surjective and the other is not.
Assuming a constant function is never injective. On a one-element domain it is — vacuously, since there are no distinct inputs.
Forgetting that id is its own inverse, which makes it the standard witness in "is this relation reflexive" style arguments.

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Identity, constant & inclusion functions — frequently asked questions

Why is the identity important?

It is the unit for composition, so it defines what an inverse means: g is inverse to f exactly when both composites equal the relevant identity.

Is a constant function ever bijective?

Only when both the domain and codomain have exactly one element. Then it is trivially the identity on that set.

What is the point of inclusion maps?

They make "X is a subset of A" into a morphism, so subset relationships can be composed and reasoned about with the same machinery as any other function.

The rest of Functions

Injective, surjective, bijective, composition, inverse. Each subtopic below has its own method, worked example and mark-losing traps.

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