evinda
Gold Member
MHB
- 3,741
- 0
Hello! (Wave)
A relation R on a set A is called strict total order if it is total order and satisfies the trichotomous identity, i.e. if R is:- antireflexive, so (\forall x \in A) \lnot(xRx) or (\forall x \in A) <x,x> \notin R
- antsymmetric, so (\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx) or (\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)
- transitive, so (\forall x \in A) (\forall y \in A) (\forall z \in A)(x Ry \wedge yRz \rightarrow xRz)
- (\forall x \in A)(\forall y \in A) \rightarrow xRy \lor yRx \lor x=yAccording to my notes, the relation of total order at the real numbers is a relation of strict total order.\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}But, in this case it will never be x=y, right? (Thinking)
A relation R on a set A is called strict total order if it is total order and satisfies the trichotomous identity, i.e. if R is:- antireflexive, so (\forall x \in A) \lnot(xRx) or (\forall x \in A) <x,x> \notin R
- antsymmetric, so (\forall x \in A)(\forall y \in A) (xRy \rightarrow \lnot yRx) or (\forall x \in A) (\forall y \in A) (<x,y> \in R \rightarrow <y,x> \notin R)
- transitive, so (\forall x \in A) (\forall y \in A) (\forall z \in A)(x Ry \wedge yRz \rightarrow xRz)
- (\forall x \in A)(\forall y \in A) \rightarrow xRy \lor yRx \lor x=yAccording to my notes, the relation of total order at the real numbers is a relation of strict total order.\langle_{\mathbb{R}}=\{\langle x,y \rangle \in \mathbb{R}^2: \text{ x is smaller that y}\}But, in this case it will never be x=y, right? (Thinking)