Qi Yining talks to someone aboutmath iff they don't talk to themselves about math to herself about math Does Yining talk Q 013 EIN True F does f represent a well defined function Q Yes a a b a o e c o d o f Ic lb floc x c 112 not defined at 0 No
1 se IDefI A set is an unordered collection of objects I 3 23 I 2,3 A E.IO the empty set of 3 O I 2 3 IN I I O I 2 Bq l p g EE g to the set of real numbers RI a c A element of the set A is an Notation means a a 4 A a is not an element of the set A means O C IN EI 040 04 IN them Russell'sParadox are not members of Let R be the set of all sets that themselves Does R exist Ex 1 4 Let R 3 R RE R R R R RER 7 R 4 12 R doesn't exist RE R so
I 1 Comparing sets IDefI The set A is a Set of B written A EB if for each a c A we have a C B and AFI ACB A EB means B B FA a c N P Fa to 4 EI.IO etNJcZcQcR EX 0 1 O l C E O I C112 e IDefI Two sets A B are equate it A EB and B E A p Q means p Q P Recall Q New Sets from Old 1 2 4 3 P S 3 e g S I 33 1 DefI Given a set 5 the power settots is the set of all subsets of S is the ordered collection An orderedn tuple cai.az an as its first element Az as its second element and has a an as its nth element
IDefI The fartesianproduotA.li An is and I a c Ai an C An ai 1,23 A EI the real plane R2 EI px px B 3,4 A X B 1,3 Ct 4 1 3 Set Operations 2,33 2,4 x c A or UNION AUB X E B x x IKEA and intersection An B act 133 r them AandB are disjoin if A 713 0 a 4 B Six IKEA and A B difference tfencotte wea often use Alb to Remi If x c Ul ACU A A 3 complement x TEY WA T
EB 1007519 EI Prove ATB TA n 13 We will show AUBT E TA n B PI XE ATB Let AUB x that is X 4 B A and Thus I and So KEI XE 13 X E F h 13 Thus We will show Tt n B E AUT C Tt n 13 Let KE Tt and a C B that is X A and B so x X A OB Then x ATB Thus XE D 2 Functions IDefI Let A B be nonempty sets A function ffromAtoB_ written f A B is an assignment of exactly one element of B to each element of A o A animal furniture3 dog B Eng f dog animal IN f IN such that f X t I x a computer program
we say A is the domain of f IDefI If f A B and B is ain of f the codom B and f IDefI Let g A c befunctions The composition B denoted fog a function from A to C defined fan dg is f gla fog by a a f glad stat A 7 C 711333 f og 2 1 Injection and Surjection tf card a 1 92 Stat B is injective or one to one if IDefI A function f A f EA f de a for all ai a az a EI injection I Ig I K f IN Z f 124 such that injection x such that f Cx I not injection IN 124 f n 21 such that fix not infection f 124 I is surjective or onto if for any IDefI A function f A B b C B an element a C A there is f S t b a
o I EI surjection 21 such that f x not surjection fi IN 104 IN such that the f 134 Surjection I 21 Such that f x I f 104 not a surjection Reme To show f is injective show V ai.az C A flail Stay a ok To show f is surjective show V be13 find a EA s t Sta b analogy A labelled balls B labelled bins f person 2.2 Bijections one correspondence if IDefI f A B is a bijection or one to is both injective and surjective f unction in A A bijection identityf EI Rt where Stx f R e bijection R where fix f Rt log x tion off IDef.I Let f B be a bijection The inversefunc A t b denoted f is defined by f B A a when Sta b f f i A co tf BB of A a
f l Let f A B be a bijection Rem B Afp B A f of Lei i Rn with fide f l y f R EI Y f i R with f x 11 f R x y y l Rt with f x ex f Rl f y lag Boy edit positive reals a bijection nonnegative reals f Rl is not Do you see why
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