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:

$\forall{\quad}n\geq{2},{\quad}\exists{\quad}primes{\quad}p,q{\quad}s.t.{\quad}p+q=2n.$

Incidentally, this statement is known as GoldbachsConjecture: it is not known whether it is true or false!

AND, OR and NOT

We can form composite statements by using and, or and not (sometimes symbolized by $\wedge$ $\vee$ and $\neg$ ), e.g.

$2>1$ and $3>2$
$5^3=100$ or $7-3=4$
not ( $4\leq{3}$ and $2+3=5$ ).

All three of these statements are true. The term or in mathematics is always inclusive, i.e. it includes the possibility of "or both", e.g. ( $1+1=2$ or $2+2=4$ ) is true.

If $P$ is a statement then not P is called the negation of P. Not P is true if and only if P is false. Note that:

not( P or Q ) is the same as ( not P ) and ( Q )
not( P and Q ) is the same as ( not P ) or ( not Q )
not( not P ) is the same as P.

Exercise. Find the negations of the following statements.

$x\leq{4};$ or $x$ is even,
$\exists$ x s.t. $x^2$ =51.

Conditional Statements

A conditional statement takes the form "if ... then ...". For example, "if Liverpool win, then Liverpool score."

This means, of course, that when the statement "Liverpool win" is true, the statement "Liverpool score" is inevitably true.

The conditional statement "if P then Q ", where P and Q are statements, can be written $P\Rightarrow{Q}$ ( $\Rightarrow$ means "implies").

In this case we also say Q if P or P only if Q. For example, $x^2=4$ if $x=2;$ but $x^2=4$ only if $x=2$ or $x=-2.$

By convention, the statement $P\Rightarrow{Q}$ is deemed to be true when P is false, e.g. both of the implications below are true(!)

$1+1=3\Rightarrow{2+2=4},$
$1+1=3\Rightarrow{2+2=5}.$

Converse and Contrapositive

Consider the implication ${P}\Rightarrow{Q}.$ The reverse implication ${Q}\Rightarrow{P}$ is called the converse, e.g. $(x=1)\Rightarrow(x^2=1)$ has converse $(x^2=1)\Rightarrow(x=1)$ Note that here the first implication is true, but the converse is false: the two statements are completely different.

The converse must not be confused with the contrapositive of $P\Rightarrow{Q},$ which is the implication $not{\quad}Q\Rightarrow{\quad}{not}{\quad}P.$ For example, "Liverpool win $\Rightarrow$ Liverpool score" has contrapositive "Liverpool don't score $\Rightarrow$ Liverpool don't win."

It is IMPORTANT to note that the contrapositive is logically identical to the original implication: they say the same thing in different ways.

Necessary and Sufficient

If $P\Rightarrow{Q}$ we say that P is sufficient for Q or that Q is necessary for P, the idea being that for Q to be true it is enough that P is true; alternatively, if P is true then Q must be true, e.g.

"For S to be a quadrilateral, it is sufficient for S to be a square."
"For S to be a square, it is necessary for S to be a quadrilateral."

If $P\Rightarrow{Q}$ and also $Q\Rightarrow{P}$ we write $P\Leftrightarrow{Q}$ and say that P is necessary and sufficient for Q, or that P if and only if Q, (sometimes abbreviated P iff Q or that P is equivalent to Q.

It is a commonly recurring theme in mathematics to show that two different-looking statements are in fact equivalent, e.g.

x is odd $\Leftrightarrow$ x+1 is even,
x+y=0 is equivalent to x=-y,
xy=0 iff x=0 or y=0,
x^2=1 if and only if x=1 or x=-1.

It is IMPORTANT to remember that an "if and only if" statement is really two separate statements.

Types of Statement

Mathematicians deal in Axioms, Definitions, Theorems and Lemmas.

Axioms

An /axiom/ is a foundational statement that we simply decree is true, e.g.

Axiom 1. $\forall$ $x,$ $x+0=x.$

Definitions

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

Definition 1. An odd number is an integer of the form 2n+1, where n is an integer.
Definition 2. An even number is an integer of the form 2n, where n is an integer.

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.

Theorem 1. The sum of an even number of odd numbers is even.

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.

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