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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:
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.
Throughout this document all variables are understood to refer to /integers:/
$\ldots{\quad},2,{\quad}1,{\quad}0,{\quad}1,{\quad}2,{\quad}\ldots$
In common language we make statements asserting that something exists or that all things of a certain type have some attribute, e.g.
All integers x satisfy $(x1)(x+1)=x^21.$
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 rewritten as follows:
 and 

We can form composite statements by using and, or and not (sometimes symbolized by $\wedge$ $\vee$ and $\neg$ ), e.g.
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:
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(!)
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.
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.
It is a commonly recurring theme in mathematics to show that two differentlooking statements are in fact equivalent, e.g.
Mathematicians deal in Axioms, Definitions, Theorems and Lemmas.
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