Transcription of Commonly Used Mathematical Notation - Columbia …
1 Commonly Used MathematicalNotation1 Logical StatementsCommon symbols for logical statement:_logical disjunction:"or"Note:in mathematics this is always an "inclusive or" "onorthe otheror both"^logical conjunction:"and":logical negation:"not"!material implication:implies; if .. thenNote:P!Qmeans:ifPis true thenQis also true;ifPis false then nothing is said aboutQcan also be expressed as:ifPthenQPimpliesQQ, ifPPonly ifQPis a su cient condition forQQis a necessary condition forPsometimes writen as)f:X!Yfunction arrow:functionfmaps the setXinto the setY function composition:f gfunction such that(f g)(x) =f(g(x))$material equivalence:if and only if (i )1 Note:P$Qmeans:meansPis true ifQis true andPis false ifQis falsecan also be expressed as:P;if and only ifQQ, if and only ifPPis a necessary and su cient condition forQQis a necessary and su cient condition forPsometimes writen as, is much less than is much greater than)therefore8universal quanti cation:for all/any/each9existential quanti cation:there exists9!
2 Uniqueness quanti cation:there exists exactly one de nition:is de ned asNote:sometimes writen as:=22 Set NotationA set is some collection of objects. The objects contained in a set are known aselements or members. This can be anything from numbers, people, other sets ,etc. Some examples of common set Notation :f;gset brackets:the set of .. ; b; cgmeans the set consisting ofa,b, andcfjgset builder Notation :the set of .. such that .. (x)gmeans the set of allxfor whichP(x)is :n2<20g=f0;1;2;3;4gNote:fjgandf:gare equivalent Notation ;empty set with no equivalent notation2set membership:is an element of=2is not an element Set OperationsCommonly used operations on sets :[UnionA[Bset containing all elements [B=fxjx2A_x2Bg\IntersectA\Bset containing all those elements thatAandBhave in common3A\B=fxjx2A^x2 BgnDi erenceorComplimentAnBset containing all those elements ofAthat are not inBAnB=fxjx2A^x =2Bg SubsetA Bsubset: every element ofAis also element ofBA Bproper subset:A BbutA6=B.]]]
3 SupersetA Bevery element ofBis also element BA BbutA6= Number SetsMost Commonly used sets of numbers:PPrime NumbersSet of all numbers only divisible by1and ;2;3;5;7;11;13;17:::gNNatural NumbersSet of all positive or sometimes all non-negative intigersN=f1;2;3; :::g, or sometimesN=f0;1;2;3; :::gZIntigersSet of all integers whether positive, negative or :::; 2; 1;0;1;2; :::g:4 QRational NumbersSet of all fractionsRReal NumbersSet of all rational numbers and all irrational numbers( numbers which cannot be rewritten as fractions, such as ,e, andp2).Some variations:R+All positive real numbersR All positive real numbersR2 Two dimensionalRspaceRnNdimensionalRspaceCCo mplex NumbersSet of all number of the form:a+biwhere:aandbare real numbers, andiis the imaginary unit, with the propertyi2= 1 Note:P N Z Q R C5
