Right-distributive


Definition

Let \((S, *, +)\) be an algebraic structure. Then \(*\) is right-distributive over \(+\) if and only if

$$\forall x,y,z \in S : (y + z) * x = (y * x) + (z * x)$$
THEOREM   

Exponentiation is right-distributive over multiplication.

$$\forall x,y,z \in \mathbb{N} : (xy)^z = x^zy^z$$

Proof available

THEOREM   

Scalar multiplication is right-distributive over vector addition.

$$(\mathbf{u} + \mathbf{v})c = \mathbf{u}c + \mathbf{v}c$$

Proof available

THEOREM   

The right operation is left-distributive over every operation.

$$x \rightarrow (y \circ z) = (x \rightarrow y) \circ (x \rightarrow z)$$

Proof available

THEOREM   

Conjunction is right-distributive over disjunction.

$$(Q \vee R) \wedge P \dashv\vdash Q \wedge P \vee R \wedge P$$

Proof available

THEOREM   

Disjunction is right-distributive over conjunction.

Proof available

THEOREM   

The left operation is right-distributive over every operation.

$$(y \circ z) \leftarrow x = (y \leftarrow x) \circ (z \leftarrow x)$$

Proof available

Proper supersets