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! 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. * EQN:1+5>2+3, * EQN:5^2+12^2=3^2, * EQN:(-1)^3=1, * EQN: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/ EQN:x,y,\ldots, etc., e.g. EQN: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 EQN:x,y,\ldots". Throughout this document all variables are understood to refer to /integers:/ EQN:\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 EQN:x satisfying EQN:x^2=49. All integers /x/ satisfy EQN:(x-1)(x+1)=x^2-1. Three useful abbreviations are s.t. (such that) and the /quantifiers:/ EQN:\exists (there exists) and EQN:\forall (for all). Thus the two previous statements could be re-written as follows: COLUMN_START [[[ EQN:\exists{x} s.t. EQN:x^2=49 ]]] COLUMN_SPLIT and COLUMN_SPLIT [[[ EQN:\forall{x;}\quad{(x-1)(x+1)=x^2-1} ]]] COLUMN_END the context here dictating that /x/ is integer. EQN:\exists and EQN:\forall can appear together - for example: * EQN:\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 EQN:\wedge EQN:\vee and EQN:\neg ), e.g. * EQN:2>1 /and/ EQN:3>2 * EQN:5^3=100 /or/ EQN:7-3=4 * /not/ ( EQN:4\leq{3} /and/ EQN: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. ( EQN:1+1=2 /or/ EQN:2+2=4 ) is true. If EQN: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. * EQN:x\leq{4}; /or/ EQN:x is even, * EQN:\exists /x/ s.t. EQN: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 EQN:P\Rightarrow{Q} ( EQN:\Rightarrow means "implies"). In this case we also say !* /Q/ if /P/ !* or !* /P/ only if /Q./ !* For example, EQN:x^2=4 if EQN:x=2; but EQN:x^2=4 only if EQN:x=2 /or/ EQN:x=-2. By convention, the statement EQN:P\Rightarrow{Q} is deemed to be /true/ when /P/ is /false,/ e.g. both of the implications below are true(!) * EQN:1+1=3\Rightarrow{2+2=4}, * EQN:1+1=3\Rightarrow{2+2=5}. ---- !! Converse and Contrapositive Consider the implication EQN:{P}\Rightarrow{Q}. The reverse implication EQN:{Q}\Rightarrow{P} is called the /converse,/ e.g. EQN:(x=1)\Rightarrow(x^2=1) has converse EQN:(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 EQN:P\Rightarrow{Q}, which is the implication EQN:not{\quad}Q\Rightarrow{\quad}{not}{\quad}P. For example, "Liverpool win EQN:\Rightarrow Liverpool score" has contrapositive "Liverpool don't score EQN:\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 EQN: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 EQN:P\Rightarrow{Q} and also EQN:Q\Rightarrow{P} we write EQN: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 EQN:\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. EQN:\forall EQN:x, EQN: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. ---- CategoryMaths