TECHNICAL LANGUAGE

A statement is an assertion which is either true or false, e.g. "London is the capital of Britain" and "The Earth is hotter than the Sun" are statements: the one true, the other false. On the other hand, the following are not statements:

• How old are you?
• Wake up!
• Capital the Britain London of is,
they are, respectively, a question, a command and nonsense.

Mathematics is concerned with statements expressed in precise and technical terms: indeed we can speak of a mathematical language with its own syntax; e.g.

• $1+5>2+3,$
• $5^2+12^2=3^2,$
• $(-1)^3=1,$
• $2(3+4=(2+3)+4.$
Of these, only the first three are statements; the fourth is nonsense (although it is expressed using mathematical symbols, it does not obey the rules of syntax). Often we deal with assertions containing variables $x,y,\ldots,$ etc., e.g. $x\geq10,x^2-y^3=z^2,-1<xy<1.$ These become statements if the variables are replaced by particular and appropriate mathematical entities in this case numbers. We shall say that these are "statements about $x,y,\ldots".$

Throughout this document all variables are understood to refer to /integers:/

$\ldots{\quad},-2,{\quad}-1,{\quad}0,{\quad}1,{\quad}2,{\quad}\ldots$

"There exists" and "For all"

In common language we make statements asserting that something exists or that all things of a certain type have some attribute, e.g.

• "There is a man in Britain over 100 years old"
• "All women in Britain are less than 200 years old"
These notions also arise in mathematical statements, e.g. There is an integer $x$ satisfying $x^2=49.$

All integers x satisfy $(x-1)(x+1)=x^2-1.$

Three useful abbreviations are s.t. (such that) and the quantifiers: $\exists$ (there exists) and $\forall$ (for all). Thus the two previous statements could be re-written as follows:
 $\exists{x}$ s.t. $x^2=49$
and
 $\forall{x;}\quad{(x-1)(x+1)=x^2-1}$
the context here dictating that x is integer. $\exists$ and $\forall$ can appear together - for example:

Definitions

A definition is a statement which states precisely what a technical term is to mean, e.g.

Theorem

A /theorem/ is any statement which is a consequence of our axioms, definitions and logical rules (though we often reserve the term for important statements), e.g.

Lemma

A lemma is a minor theorem which is a key step to proving a more major theorem.

• Lemma 1. The sum of two odd numbers is even.

CategoryMaths