Injection, surjection, bijection
Functions from one set to another: image, preimage, injective, surjective and bijective functions, the inverse function, and what the cardinalities of finite sets allow.
- 27 cards
- 4 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
- Sort and link
- The numbers
A few cards from the deck
Image and preimage: what is it?
Show the answer
If f(x) = y, then y is the image of x and x is a preimage of y. Every x has exactly one image, but a y may have zero, one or several preimages.
For x ↦ x² on ℝ, 4 has two preimages (2 and −2), 0 has one, −1 has none. The image is unique, the preimages are not: injective and surjective say what happens.
Let f: ℝ → ℝ, x ↦ x². What is the preimage of {4}?
- {−2, 2}
- {2}
- {16}
- {4}
Show the answer
{−2, 2}
The preimage of a set B is the set of all x such that f(x) is in B: x² = 4 gives x = 2 or x = −2. Do not confuse it with the direct image of {4}, which is {16}.
A function f is injective when f(x) = f(y) implies …: two distinct elements have … images.
Show the answer
A function f is injective when f(x) = f(y) implies x = y: two distinct elements have distinct images.
It is the contrapositive of “x ≠ y implies f(x) ≠ f(y)”: both forms are equivalent, and you pick the more convenient one for a proof.
Injective function: what is it?
Show the answer
A function in which two distinct elements always have distinct images: every element of the target has at most one preimage.
To prove it, assume f(x) = f(y) and show x = y. To refute it, one counterexample is enough: f(−2) = f(2) for x ↦ x².