LINEAR ALGEBRA (MATH 317H) 15 4. Mathematical theorems, some formal logic 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 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 () = 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. Collections: Mathematics