Do claims in economics require proofs? My professor stated four axioms in class 


Scarcity
Rationality (aka purposive behavior) 
Stable Preferences 
Equilibrium 


I don't understand what these axioms are for. I am used to axioms with distinct mathematical properties (eg definition of a vector space) and these axioms don't have that. 
I had an exam question that said 


  True/False/Uncertain. The Law of Demand is a direct result of diminishing marginal utility. 


and I was supposed to use the scarcity axiom. But what does it mean to show this claim is false? The principle of scarcity is just that "resources are scarce". That's not mathematical. How can I prove the above statement is false using it? 

My Question:

How can I tell when a question requires a proof versus merely conceptual reasoning / explaining? What is the difference between the two?
These assumptions actually have distinct mathematical ground, but your professor likely decided to leave this out for (of course bad) pedagogic reasons.


scarcity: society has insufficient productive resources to fulfil all human wants and needs. Scarcity is the motive behind economics. If it is cheaper for everybody to get a given good from the ground than to produce it, then they are not economic goods, they don't have a market value and are not treated by introduction classes in economics. This can be stated as: for all actor, for all (studied) goods, the cost of acquiring the good is strictly positive.
Rationality: people adopt the best actions to achieve their goals. Usually, this is translated in maths by introducing a utility function from the possession set to a measuring set (usually $\mathbb{R}$), and stating people always act in order to maximize their utility.
Stable Preferences: Using the same utility function as above, this states that this function does not depend of time. This implies that if you prefer having a kilogram of soup than a kilogram of raw potatoes today, it will still be the case tomorrow.
Equilibrium: Here I have to guess, but given your example I suppose you're studying equilibrium on the market. I guess what your teacher meant is that you study the economy at a time when the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers.


All these definitions are not guaranteed to match with those implied by your teacher, but are consistent and widely taught in economics. When not otherwise specified, economical answers require referring to a consistent framework of axioms from which mathematics should be able to guide you to the solution you're trying to state. Different models can show different solution, but one model should give only one solution (which can be composite: "people buy more or less goods at the end of the month" is a valid solution, but "people buy more goods at the end of the month" and "people buy less goods at the end of the month" should not be obtained from the same data with the same model...).

However, economic practice does not require you to re-demonstrate any known result (mostly, the results which you know a name for can be used as starting points for your answers). If you're not confident with "explaining", you need to be able to think about a one-line mathematical justification for each explanation you give, and thus you will ensure your explanations give you the expected marks.