Lecture - Introduction to University Mathematics, V

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 $Y$?

\[|X| = |Y|\]