Science Metro

二項関係と写像

$二項関係$

R

が

A

と

B

間の二項関係 :

R \subset A \times B

$A上の二項関係$

R

が

A

上の二項関係 :

R \subset A \times A

$写像$

A

から

B

への写像

G_{f}

:

1.\forall a \in A \exists b \in B ((a, b) \in G_{f}) 2. (a, b_{1}) \in G_{f} \land (a, b_{2}) \in G_{f} \Rightarrow b_{1} = b_{2}

f : A \to B (x \mapsto f (x)) :=

G_{f}

f (x) = y def ⟺ (x, y) \in G_{f}

f = g def ⟺ G_{f} = G_{g}

$全射$

f

が全射:

\forall b \in Y \exists a \in X (f (a) = b)

$単射$

f

が単射:

\forall a_{1}, a_{2} \in X (f (a_{1}) = f (a_{2}) \to a_{1} = a_{2})

$全単射$

f

が全単射:

f

が全射かつ

f

が単射

$恒等写像$

X

上の恒等写像:

G_{i d_{X}} = {(x, y) \in X \times X ∣ y = x}

$逆写像$

f

が全単射の時

G_{f^{- 1}} := {(y, x) \in Y \times X ∣ f (x) = y}

$命題$

f

は全単射

\Leftrightarrow \exists g : B \to A (g \circ f = i d_{X} \land f \circ g = i d_{Y})

$合成写像$

f, g

の合成写像

G_{g \circ f} := {(x, z) \in X \times Z ∣ z = g (f (x))}

$命題$

f : X \to Y

g : Y \to Z

に対して次が成り立つ

1.

f, g

全射

\Rightarrow g \circ f

は全射

2.

f, g

単射

\Rightarrow g \circ f

は単射

$合成写像の演算法則$

h \circ (g \circ f) = (h \circ g) \circ f

(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}

$像$

f

の

X_{0}

の像:

f (X_{0}) := {y \in Y ∣\exists x \in X_{0} (f (x) = y)}

$定理$

(1) X_{1} \subset X_{2} \Rightarrow f (X_{1}) \subset f (X_{2})

(2) f (X_{1} \cup X_{2}) = f (X_{1}) \cup f (X_{2})

(3) f (X_{1} \cap X_{2}) \subset f (X_{1}) \cap f (X_{2})

$逆像$

f

の

Y_{0}

の逆像:

f^{- 1} (Y_{0}) := {x \in X ∣ f (x) \in Y_{0}}

$定理$

(1) Y_{1} \subset Y_{2} \Rightarrow f^{- 1} (Y_{1}) \subset f^{- 1} (Y_{2})

(2) f^{- 1} (Y_{1} \cup Y_{2}) = f^{- 1} (Y_{1}) \cup f^{- 1} (Y_{2})

(3) f^{- 1} (Y_{1} \cap Y_{2}) = f^{- 1} (Y_{1}) \cap f^{- 1} (Y_{2})