 
Summary: LINEAR ALGEBRA (MATH 317H) 15
Lecture 3
4. Mathematical theorems, some formal logic
A Mathematical theorem is the statement of a fact. A proof of a theorem is a
logical explanation why the statement is true. Note that in mathematics a state
ment is either true or false, there is nothing in between. The statement 'the leaves
of the maple tree are green' becomes false if the tree has one single nongreen leaf.
A theorem usually consists of a hypothesis and a conclusion. It is important that
you can identify the hypothesis (what is being assumed) and the conclusion (what
has to be proved) in a theorem. Here is an example from Calculus:
Theorem: Let f be a differentiable real valued function defined on R. Let [a, b]
be some closed interval. Then there exists [a, b] such that
f
() =
f(b)  f(a)
b  a
.
. The hypothesis of the theorem, i.e. what can be assumed for the proof is
· f is a differentiable real valued function defined on R.
· [a, b] is a closed interval.
