Foundations of Maths I

Definition 3.9

What makes functions $f: X → Y$ and $g: A → B$ equal?

Two functions $f: X \rightarrow Y$ and $g: A \rightarrow B$ are equal if and only if:

1) $X = A$

2) $Y = B$

3) $f(x) = g(x)$ for all $x \in X$

What does it mean for a function $f$ to be injective?

$$f: X \rightarrow Y$$

f is injective if different elements in X are mapped to different elements in Y, so whenever $f(x_1) = f(x_2)$ for $x_1, x_2 \in X$, then $x_1 = x_2$

What does it mean for a function $f$ to be surjective?

$$f:X \rightarrow Y$$

$f$ is surjective if any $y$ in $Y$ is the image of some element $x$ in $X$, that is $\forall y \in Y \exists x \in X: y=f(x)$

Definition 4.1

What is a binary Operation?

A binary operation on a set G is a function $\ast: G \times G \rightarrow G$

Definition 3.22

Describe the Equivalence classes