Negative transitivity

A third element z is placed in an existing binary relation between x and y, so that negative transitivity is fulfilled.

The negative transitivity of a two-digit relation on a set is given if: ${\displaystyle R}$

${\displaystyle \forall x,y,z:\neg xRy\land \neg yRz\Rightarrow \neg xRz.}$

Strictly weak orders fulfill negative transitivity.

Equivalent transformations

Sometimes the connection of negative transitivity is formulated as follows:

${\displaystyle (x>y)\Rightarrow (x>z)\lor (z>y)}$

This representation is obtained by negating an implication . If the expressions in brackets are replaced by the general statements A, B and C, it follows:

${\displaystyle \neg A\land \neg B\Rightarrow \neg C}$

If this expression is negated again, firstly the direction of implication is reversed and secondly, according to De Morgan's laws, both the negations of A and B are canceled, but the conjunction is also transformed into a disjunction:

${\displaystyle \neg (\neg A\land \neg B)\Leftarrow C}$

${\displaystyle A\lor B\Leftarrow C}$

This then basically corresponds to the form we started from above.

Example in everyday language

If milk costs no less than bread and bread costs no less than cake, then milk costs no less than cake either.

Microeconomics

In microeconomic household theory , negative transitivity and asymmetry are used as assumptions for the strict preference relation .

See also

• Transitive relation

Individual evidence

1. ^ Friedrich Breyer: Microeconomics: An introduction . Jumper; Edition: 5 ed. 2011 (September 25, 2011). ISBN 978-3642221491 . Page 166.
2. ^ Friedrich Breyer: Microeconomics: An introduction . Jumper; Edition: 5 ed. 2011 (September 25, 2011). ISBN 978-3642221491 . Page 166.