Property vs Definition What's the difference between classifying a mathematical statement as a definition and a property? From what I've read so far, the mathematical statements written as property or definition are almost identical. So it confuses me when is a mathematical statement a definition versus a property? And does property need proving?
 A: A definition is something the author gets to decide — more or less. There are canonical ways to define many things, but there are sometimes competing and non-equivalent definitions of terms.
For example, if I am writing an analysis book and want to define what an "increasing function" is, I might write that it means that if $x<y$ then $f(x)\leq (y)$. Another author might chose to write $f(x)<f(y)$ (with strict equality). In either case, as long as we are consistent throughout our text books in what we mean by "increasing," it is okay that the definitions are different.
"Property" is a more vague word. It usually follows from a definition. Furthermore, a property usually describes special classes of things.
For example, one property of increasing functions (on closed intervals) is that they are integrable. We could also state this fact as a theorem.

As another way to think about it, definitions are almost always a way to correlate english words with mathematical statements. We want to know how to interpret the intuitive english phrase "increasing functions" with a rigorous mathematical statement.
A: Maybe you mean these two ways to define a set?
$$
S = \{ 2,3, 4, 5 \}
$$
Here we listed the elements explicitly.
Then we have this kind of definition:
$$
S = \{ n \in \mathbb{N} \mid n \bmod 2 = 1 \}
$$
This definition of the set $S$ selects elements from $\mathbb{N}$ by the property $n \bmod 2 = 1$ of these elements. Only those elements of $\mathbb{N}$ where that property is true for are part of $S$.
A: From Wikipedia:
A definition is defined as:

In mathematics, definitions are generally not used to describe existing terms, but to give meaning to a new term.

A property is defined as:

In math terminology, a property $p$ defined for all elements of a set $X$ is usually defined as a function $p: X \to \{\text {true, false}\}$, that is true whenever the property holds; or equivalently, as the subset of $X$ for which $p$ holds; i.e. the set $\{x| p(x) = \text {true}\}; p$ is its indicator function.

A: It seems to me if the author states by definition, he is asserting a fact that you need to take without proof. properties, on the other hand, are either direct consequences of definitions ( no proof necessary) or complex consequences of definitions and other properties that can be proven.
"Set operations have several properties, which are elementary consequences of the definitions " introduction to probabilities 2nd edition, Bertsekas and Tsitsiklis.
