Summary: LINEAR ALGEBRA (MATH 317H) 15
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 non-green 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(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.