Anti-symmetric relation


Definition

A binary relation \(\mathrel{R}\), on a set \(S\), is anti-symmetric if and only if:

$$\forall x,y \in S : x{\mathrel{R}}y \wedge y{\mathrel{R}}x \Rightarrow x = y$$
THEOREM   

The less than or equal to relation is anti-symmetric.

Proof unavailable

THEOREM   

Let \((S, \sqcup, \preceq)\) be a meet semilattice. Then \(\preceq\) is antisymmetric.

$$\forall a,b \in S : a \preceq b \wedge b \preceq a \Rightarrow a = b$$

Proof available

THEOREM   

The subset relation is antisymmetric.

$$A \subseteq B \wedge B \subseteq A \Rightarrow A = B$$

Proof available

THEOREM   

Let \((S, \sqcup, \preceq)\) be a join semilattice. Then \(\preceq\) is antisymmetric.

$$\forall a,b \in S : a \preceq b \wedge b \preceq a \Rightarrow a = b$$

Proof available

Proper supersets