Science Metro

Ordered set

$Partial order$

R

is a partial order of

A

:

\forall a, b, c \in A

1. a \leq_{R} a 2. (a \leq_{R} b \land b \leq_{R} c) \to a \leq_{R} c 3. (a \leq_{R} b \land b \leq_{R} a) \to a = b

$＜$

a

＜

_{R} b : (a \leq_{R} b) \land (a \neq = b)

$Total order$

R

is a total order on

A

4.

\forall a, b \in A (a \leq_{R} b \lor b \leq_{R} a)

$Ordered set$

(A, \leq_{R})

is an ordered set:

R

is a partial order on

A

$Totally ordered set$

(A, \leq_{R})

is a totally ordered set:

R

is total order on

A

$Upper bound$

x

is an upper bound of

A

:

\forall y \in A (y \leq_{R} x)

$Under bound$

x

is an under bound of

A

:

\forall y \in A (x \leq_{R} y)

$Maximum element$

x

ia a maximum element of

A

:

x \in A \land x

is an upper bound of

A

$Minimum element$

x

is minimum element of

A

:

x \in A \land x

is an under bound of

A

$Corollary$

Maximum and Minimum elements are unique if they exist

$Supremum$

Expected $

$Infimum$

x

is an infimum of

A

in f A

:

maximum of {y \in P ∣ y is an under bound of A}

$Corollary$

Supremum and infimum elements are unique if they exist

$Corollary$

Supremum is an upper bound

Infimum is an under bound

$Corollary$

ma x A

exists

\Rightarrow ma x A = s u p A

min A

exists

\Rightarrow min A = in f A

$Maximal element$

x

is a maximal element

A

:

x \in A \land \forall y \in A (x \leq_{R} y \to x = y)

$Minimal element$

x

is a Minimal element of

A

:

x \in A \land \forall y \in A (y \leq_{R} x \to x = y)

$Corollary$

max A

exists

\Rightarrow max A

is maximal element of

A

min A

exists

\Rightarrow min A

is minimal element of

A