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

Definition 43 The unary product operation can be defined by the equations o i=o ni = s(o)
and s(m) i=o ni = ns(m) ⊙ m i=o ni.
c○: Michael Kohlhase 40
2.2.2 Talking (and writing) about Mathematics
Before we go on, we need to learn how to talk and write about mathematics in a succinct way.
This will ease our task of understanding a lot.
Talking about Mathematics (MathTalk)
Definition 44 Mathematicians use a stylized language that
uses formulae to represent mathematical objects, 2 e.g.
0 3
x 2 dx 1
uses math idioms for special situations (e.g. iff, hence, let. . . be. . . , then. . . )
classifies statements by role (e.g. Definition, Lemma, Theorem, Proof, Example)
We call this language mathematical vernacular.
Definition 45 Abbreviations for Mathematical statements
∧ and “∨” are common notations for “and” and “or”
“not” is in mathematical statements often denoted with ¬
∀x.P (∀x ∈ S.P ) stands for “condition P holds for all x (in S)”
∃x.P (∃x ∈ S.P ) stands for “there exists an x (in S) such that proposition P holds”
∃x.P ( ∃x ∈ S.P ) stands for “there exists no x (in S) such that proposition P holds”
∃ 1 x.P (∃ 1 x ∈ S.P ) stands for “there exists one and only one x (in S) such that proposition
P holds”
“iff” as abbreviation for “if and only if”, symbolized by “⇔”
the symbol “⇒” is used a as shortcut for “implies”
Observation: With these abbreviations we can use formulae for statements.
Example 46 ∀x.∃y.x = y ⇔ ¬(x = y) reads
“For all x, there is a y, such that x = y, iff (if and only if) it is not the case that
x = y.”
c○: Michael Kohlhase 41
b EdNote: think about how to reactivate this example
We will use mathematical vernacular throughout the remainder of the notes. The abbreviations
will mostly be used in informal communication situations. Many mathematicians consider it bad
style to use abbreviations in printed text, but approve of them as parts of formulae (see e.g.
Definition 2.2.3 for an example).
To keep mathematical formulae readable (they are bad enough as it is), we like to express mathematical
objects in single letters. Moreover, we want to choose these letters to be easy to remember;
e.g. by choosing them to remind us of the name of the object or reflect the kind of object (is it a
number or a set, . . . ). Thus the 50 (upper/lowercase) letters supplied by most alphabets are not
24