Equality to the Third

In mathematics, the binary relation $R$ ${\ displaystyle R}$ $R$ on the set $X$ ${\ displaystyle X}$ $X$ possesses the property of equality to the third if, for any three elements of the set $a, b, c$ ${\ displaystyle a, b, c}$ $a, b, c$ relationship fulfillment $a R b$ ${\ displaystyle aRb}$ $aRb$ and $c R b$ ${\ displaystyle cRb}$ ${\ displaystyle cRb}$ entails fulfillment of the relationship $a R c$ ${\ displaystyle aRc}$ $aRc$ .

Formally, the attitude $R$ ${\ displaystyle R}$ $R$ has the property of equality to the third if $\forall a,b,c\in X,\ aRb\land cRb\Rightarrow aRc$ ${\ displaystyle \ forall a, b, c \ in X, \ aRb \ land cRb \ Rightarrow aRc}$ ${\ displaystyle \ forall a, b, c \ in X, \ aRb \ land cRb \ Rightarrow aRc}$ .

If the relation is symmetric , then the property of equality to the third coincides with transitivity .

Example

All transitive and symmetric relations have this property.

Equality to the Third

Example

See also

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