General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

24 CHAPTER 3. ELEMENTARY DISCRETE MATH
Definition 26 A theorem is a statement about mathematical objects that we know to be
true.
We reason about mathematical objects by inferring theorems from axioms or other theorems,
e.g.
1. “ ” is a unary natural number (axiom P1)
2. / is a unary natural number (axiom P2 and 1.)
3. // is a unary natural number (axiom P2 and 2.)
4. /// is a unary natural number (axiom P2 and 3.)
Definition 27 We call a sequence of inferences a derivation or a proof (of the last statement).
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If we want to be more precise about these (important) notions, we can define them as follows:
Definition 28 In general, a axiom or postulate is a starting point in logical reasoning with the
aim to prove a mathematical statement or conjecture. A conjecture that is proven is called a
theorem. In addition, there are two subtypes of theorems. The lemma is an intermediate theorem
that serves as part of a proof of a larger theorem. The corollary is a theorem that follows directly
from another theorem. A logical system consists of axioms and rules that allow inference, i.e. that
allow to form new formal statements out of already proven ones. So, a proof of a conjecture starts
from the axioms that are transformed via the rules of inference until the conjecture is derived.
We will now practice this reasoning on a couple of examples. Note that we also use them to
introduce the inference system (see Definition 28) of mathematics via these example proofs.
Here are some theorems you may want to prove for practice. The proofs are relatively simple.
Let’s practice derivations and proofs
Example 29 //////////// is a unary natural number
Theorem 30 /// is a different unary natural number than //.
Theorem 31 ///// is a different unary natural number than //.
Theorem 32 There is a unary natural number of which /// is the successor
Theorem 33 There are at least 7 unary natural numbers.
Theorem 34 Every unary natural number is either zero or the successor of a unary natural
number. (we will come back to this later)
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Induction for unary natural numbers
Theorem 35 Every unary natural number is either zero or the successor of a unary natural
number.
Proof: We make use of the induction axiom P5:
P.1 We use the property P of “being zero or a successor” and prove the statement by
convincing ourselves of the prerequisites of
P.2 ‘ ’ is zero, so ‘ ’ is “zero or a successor”.