集合の要素の定義

$X$: a set
" $x \in X$ " means $x$ is a element of $X$

部分集合の定義

$A$: a set
$B$: a set
$$A \subset B \overset{\text{def}}{\Longleftrightarrow} {}^\forall a \in A { \text , } a \in B$$ $$A \supset B \overset{\text{def}}{\Longleftrightarrow} B \subset A$$

冪集合の定義

$$P(X) := \{ A | A \subset X \}$$ $P(X)$ called a power set.

写像の定義

$X$: a set
$Y$: a set
$$\forall x \in X, f(x) \in Y$$ when $f(x)$ Uniquely determined, then f called mapping(写像).

像(Image)の定義

$f:X \rightarrow Y$ is a mapping.
$A \subset X$
$B \subset Y$
$f(A):=\{ f(x) | x \in A \}$
$f^{-1}(B):=\{ x | f(x) \in B \}$
then
$f(A)$ called image of $A$ by $f$
$f^{-1}(B)$ called inverse (image) of $B$ by $f$

単射・全射の定義

Suppose $f:X \rightarrow Y$ is a mapping.
$$a,b\in X \text{ and } a\neq b \Rightarrow f(a) \neq f(b) \overset{\text{def}}{\Longleftrightarrow} f \text{ is called "Injection".}$$ $$\forall y\in Y, \exists x \in X, y = f(x) \overset{\text{def}}{\Longleftrightarrow} f \text{ is called "Surjection".}$$

順序集合の定義

$X$ is a set. A binary relation $\ge$ on $X$ fills following conditions, then $(X,\ge)$ is called a ordered set. * $\forall x \in X, x \ge x$ * $\forall x, y \in X, x \ge y$ or $y\ge x$. * $\forall x, y, z \in X, x \ge y, y \ge z \Rightarrow x \ge z$