B.2 What is a theorem?
So, a definition is essentially a predicate whose truth set consists of our objects of interest. The whole point of mathematics is to find true propositions involving those objects, most often in the form A →B. Consider the following theorem.
Theorem 144. (Existence of global minima for convex functions)
Let ![f : [0,1] → ℝ](https://static.packt-cdn.com/products/9781837027873/graphics/media/file2117.png)
be a function. If 
is continuous, then there exists an 
such that 
assumes its minimum at 
on ![[0,1]](https://static.packt-cdn.com/products/9781837027873/graphics/media/file2122.png)
.
(That is, for all ![x ∈ [0,1]](https://static.packt-cdn.com/products/9781837027873/graphics/media/file2123.png)
, we have 
.)
Don’t worry if you are unfamiliar with the concepts of continuity and minimum; it’s beside the point. The gist is that Theorem 144 can be written as
where F denotes the set of all functions [0,1] →ℝ, and the predicates C(f) and M(f) are defined by
![C (f ) : f is continuous on [0,1], ∗ ∗ M (f ) : ∃x ,∀x ∈ [0,1],f (x ) ≤ f(x).](https://static.packt-cdn.com/products/9781837027873/graphics/media/file2126.png)
Notice the structure of the theorem: “Let x ∈A. If B(x), then C(x).” With the first sentence, we are setting the domains of the predicates A(x) and B(x), and putting a universal quantifier in front of the conditional ...
