What does zeroth law mean?

Hint: Recall the zeroth law pertaining to thermodynamics. Determine how the zeroth law is used to deduce the meaning of two systems in thermal equilibrium and how this subsequently gives rise to the concept of temperature as a physical quantity that is essential in defining the state of thermal equilibrium of a system. Also determine the transitive nature of the quantities that this law entails.

Complete answer:
We know that the laws of thermodynamics define physical quantities like temperature, energy and entropy and characterize thermodynamic systems at thermodynamic equilibrium. They state empirical facts that form the basis of estimating the possibility of certain phenomena such as perpetual motion.
Traditionally, thermodynamics has recognized three fundamental laws, named as the first law, second law and the third law, in the order of their postulation in history. However, a more fundamental statement was labelled the zeroth law by R.H. Fowler in 1931, which was defined more than half a century later since the first and second laws of thermodynamics had been formulated. The term “zeroth” has been used to simply denote that it should come before the first and second laws since it lays the necessary groundwork to better understand the successive laws.
The zeroth law of thermodynamics introduces the concept of thermal equilibrium and paves way towards the definition of temperature as a physical quantity. It states that two systems in thermal equilibrium with a third system are in equilibrium with each other. At the face of it, this law appears to be simple and obvious, but it has a profound depth of meaning implying that two systems X and Y may know Z but they may not know each other and still be in equilibrium with respect to each other.
When two or more systems are in thermal equilibrium, they are said to have the same temperature. Thus, temperature is essentially a scalar quantity which is a property of all thermodynamic systems (in equilibrium) such that temperature equality is a necessary and sufficient condition for thermal equilibrium.
Thus, the zeroth law encompasses the transitive nature of thermal equilibrium.

Note:
The zeroth law is a transitive relation such that if X is in thermal equilibrium with Y and Y is in thermal equilibrium with Z, then X is in thermal equilibrium with Z. The zeroth law is also a symmetric relation, i.e., if a system X is in thermal equilibrium with Y, then Y is in thermal equilibrium with X. Additionally, the fact that every thermodynamic system is in thermal equilibrium with itself implies that the zeroth law is a reflexive relation. Thus, a reflexive, transitive relationship does not guarantee an equivalence relation but a reflexive, symmetric relation does. We can conclude that the zeroth law is basically an equivalence relation by which a thermodynamic system can be identified.