Left-cancellative


Definition 1

ProofWiki

Definition 2

Let \((S, \circ)\) be an algebraic structure. Then \(\circ\) is left-cancellative if and only if

$$\forall a,b,c \in S: a \circ b = a \circ c \Rightarrow b = c$$
THEOREM   

Natural number addition is left-cancellative.

$$\forall a,b,c \in \mathbb{N} : a + b = a + c \Rightarrow b = c$$

Proof available

LEMMA   

Integer addition is left-cancellative.

$$\forall x,y,z \in \mathbb{Z} : z + x = z + y \Rightarrow x = y$$

Proof available

Proper supersets