# Lecture - Introduction to University Mathematics, V

> Source: https://ollybritton.com/notes/uni/prelims/mt22/introduction-to-uni-maths/lectures/lecture-5/ · Updated: 2022-10-16 · Tags: uni, lecture

### Flashcards
What is the image of a function $f : X \to Y$ in set notation?::
$$
\\{f(x) : x \in X\\}
$$

What does $f(A)$ mean for $A \subseteq X$?::
The image of $A$ under the function.

What does the pre-image $f^{-1}(A)$ mean in set notation for a function $f : X \to Y$ and $A \subseteq Y$?::
$$
\\{ x \in X : f(x) \in B \\}
$$

What does $f(A)$ mean for $A \subseteq X$?::
The image of $A$ under the function.

What does the notation $f{\restriction_A}$ mean for a function $X \to Y$?::
The function with domain $A$ and range $Y$.

What does it mean ((in English)) for a function to be injective?::
One-to-one

What does it mean ((in notation)) for a function to be injective?::
$$
f(x_1) = f(x_2) \implies x_1 = x_2 
$$

What does it mean ((in English)) for a function to be surjective/onto?::
Every element in the range can be reached by an element in the domain.

What does it mean ((in notation)) for a function to be surjective/onto?::
$$
\forall y \in Y \text{ } \exists x \in X : f(x) = y
$$

What does it mean ((in English)) for a function to be bijective?::
It is injective and surjective, and so invertible.

What’s the word for a function that’s one-to-one?::
Injective.

What’s the word for a function where every element in the codomain can be reached by an element in the domain?::
Surjective/onto.

If $X$ and $Y$ are finite sets, and $f : X \to Y$ is injective, what must be true about the cardinality of $X$ and $Y$?::
$$
|Y| \ge |X|
$$

If $X$ and $Y$ are finite sets, and $f : X \to Y$ is surjective, what must be true about the cardinality of $X$ and $Y$?::
$$
|X| \ge |Y|
$$

If $X$ and $Y$ are finite sets, and $f : X \to Y$ is bijective, what must be true about the cardinality of $X$ and $Y$?::
$$
|X| = |Y|
$$

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt