Zeroth law of thermodynamics and why do we need it ?

Zeroth law of thermodynamics and why do we need it ?

The zeroth law of thermodynamics states that if two thermodynamic
systems are each in thermal equilibrium with a third, then they are in
thermal equilibrium with each other. Accordingly, thermal equilibrium
between systems is a transitive relation.

(if A =B and B=C, then C=A)

Source

The fact that temperature obeys a transitive property is by no means intuitive!

If A likes B; B likes C; Does not imply that A likes C.

The city closest to city A is B, the city closest to B is C, but A is not always the closest to C

Or this popular example.

Rock > Scissors, Scissors > Paper. But Rock > Paper is definitely not true

To remind us that temperature obeys a transitive property is the reason why there exists a zeroth law of thermodynamics.

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