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# Note for MATH 3330 at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15

Math 3330 Section 12 Mappings An ordered pair ab ab cd ifand only if ac and Ed The Cartesian Product of sets A and B vc pw 51 AgtltBabaeAbeB s a M01050 M W i s N OT A 1356 and B 27 MIWW AXE 2CSIQWIW6 Let A and B be nonempty sets A subset f of A X B is a mapping from A to B ifand onlyiffor each a E A there is aunique b E B suchthat a b E f We also Write b fa and call b the image ofa under f man The set A is called the domain off and B is called the codomain off The image of f Written fA is the set CyeByfa forsomeaeA Subsets and Mappings Let S g A and let f I A gt B be amapping Then fS y e B y fs for some s e S is calledthe image of S under f Let T g B Then f 1T x e A fx 6 T The SET f 1T is called the inverse image ofT under f Composition of Mappings Just as happens in function notation in calculus mappings can be put together by composition For g I A gt B f I B gt C the composition fog A gt C is de nedby f o ga fgafor all a e Z a Special Mappings Gk H b A Onto 7 Surjection Mappings A map f I A gt B is called surjective or onto if and only iffAB Wc is W Onetoone 7 lnjective Mappings A map f I A gt B is called injective or onetoone if and only ifdifferent elements of A always have different images under f Bijective Mappings 7 Onetoone correspondence A map f I A gt B is called bijective ifand only ifit is surjective and injective Another name for a bijective map is a onetoone correspondence between A and B Picture Representations of Mappings P 4 O ISOLlVL IS OA39l o 6 Met mvl om 1b i d gslBls A 5 NStOX VL mam Mappings are most often represented by a formula for the image of an element of A This is equivalent to functions you should be very familiar with from previous math classes The new twists will appear in later chapters when we require our maps to have other special properties Let s practice working with sets and maps Notice that the de nitions for injective and surjective maps depend on the formula for the mapping AND the sets involved We are NOT assuming that our functions are given for all real numbers only for the sets of numbers speci ed Eamples Determine if the following maps are surjective injective or bijective Q E 5 E a KM j 5 01920 fr x17X w D j o 7quot 139 7 bxjcdll d D is ane chme Fizzm warm Mg KxD 6ltQ 3 W 7936 ab 6 gt x tgtlt2l v 7951 6CX7 5 O hbs Que44441 k a 4 4 Cl XX Jy 663117 IQQX1Hi X 71 I X IX W1 1 2 2 393 70 ax a W m kf 12gtlthl QXS E xts g n3 vacfive 39 par QCLch fZ gtZfx 5 xeve 2x xodd 1 5 onJVo becqygc 75 65 c g 9 5 55 7 1 A 93 GoF W am a gm g L E 2ltLr2owggt i ga11E g gt E XJQ SFW 39 X7 6 gt6 M7tvgm W XVXgeg Wham gt 932K a x AEandBgE Ea E I gf 2 mm m om g is NOT mot W Wm 65x 2 c ampX Xf 3 M W M X65 W MM xgtgtQ a JOOEC39HU i 1 5 ca a 67 5X7 ff Lg f z giveltby fx 1 1 3 11ng eri 72gt X S 0 1 2 nd fS f 1fS 53 6714 55 3433 051 67763 50 2 6amp0 Q39Q K 3 A 8 T5 6 nd f 1Tff 1T CEDj O ng39 X71 elk 6 T 68 QEch W M W x th tb er We m Kgm gm m3 Let f I A gt B With A and B nonempty Prove that f has the property that f71f S for every subset S ofA ifand only iffis oneto one Q 0 Aism gltsgts VS 74 cwy XWYa Isaw K fgo Q M 3189 6quot3C57gtgt5 x fxg 5amp3 G 3 Vt ASIWC 60 6 g7 X26 6 gtlt EiD 37 X26 E55125 j XEEX g is ON39OAO7D FrOV r g S W Y 765 c 5 gea MAM mw 60K MW 4 S M L mw X XQ ZS

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