# Minimum and maximum

THEOREM    $$\forall a,b \in \mathbb{R} : a + b = \min(a, b) + \max(a, b)$$

Proof available

THEOREM

Minimum is left-distributive over maximum.

$$\forall a,b,c \in \mathbb{R} : \text{min}(a,\text{max}(b,c))=\text{max}(\text{min}(a,b),\text{min}(a,c))$$

Proof available

THEOREM

Minimum is right-distributive over maximum.

$$\forall a,b,c \in \mathbb{R} : \text{min}(\text{max}(b,c),a)=\text{max}(\text{min}(b,a),\text{min}(c,a))$$

Proof available

THEOREM

Maximum is left-distributive over minimum.

$$\forall a,b,c \in \mathbb{R} : \text{max}(a,\text{min}(b,c))=\text{min}(\text{max}(a,b),\text{max}(a,c))$$

Proof available

THEOREM

Maximum is right-distributive over minimum.

$$\forall a,b,c \in \mathbb{R} : \text{max}(\text{min}(b,c),a)=\text{min}(\text{max}(b,a),\text{max}(c,a))$$

Proof available

THEOREM    $$\forall x,y \in \mathbb{R} : \max(\sgn(x), \sgn(y)) = \sgn(\max(x,y))$$

Proof available

THEOREM    $$\forall x,y \in \mathbb{R} : \min(\sgn(x), \sgn(y)) = \sgn(\min(x, y))$$

Proof available