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11.96 0 0 11.96 0 0 Tm 22.258 48.189 Td (\() Tj A->10mRJ8.lFRP](G.e5T)/f7-_aBN/.dtBP1du5/k" (&K]&tBWPuZ`-aus&i60"cm'>`(4gW;uh$WVYooHM0kiDOEIdUD5pTK\>&]tc1?D2BVno/3oGI[b'X /Differences [ 32 /.notdef 34 /quotedblright /.notdef /.notdef 38 /.notdef /quoteright 42 /.notdef 60 /.notdef 62 /.notdef 89 /.notdef /.notdef 92 /quotedblleft 94 /.notdef /.notdef /quoteleft 123 /endash /emdash /.notdef /.notdef 128 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /Gamma /Delta /Theta /Lambda /.notdef /Pi /Sigma /Upsilon /Phi /Psi /.notdef /.notdef /Omega /ff /fi /fl /ffi /ffl 180 /dotlessj /grave /acute /caron /breve 187 /cedilla /germandbls /ae /oe /.notdef /.notdef /OE /Oslash /suppress /dieresis 199 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 216 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef 241 /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef ] Tg_@nC@g*0&&r!s6Kjkh/8eU7.feQMki9dqd.`PDCA,]dKE!s[ 0.798 0 Td (=) Tj Functions and relations are one the most important topics in Algebra. /ItalicAngle -14 4.452 0 Td (S) Tj /F4 1 Tf The modern study of infinity began with Georg Cantor and the birth of set theory, so we naturally start with sets. 1.178 0 Td (A) Tj A function is a special type of relation where every input has a unique output. /F5 1 Tf Other Examples of One-to-One Relationships. EbpY74Tn7(#MU_`daldf&S^`"tJW7Y#d@(Wn]MKPj\Gh)@>;7`hC&q`s$FZQ_%@M-F31D]EM[C4MGL 7.97 0 0 7.97 0 0 Tm /F5 1 Tf 0.972 0 Td (the) Tj 11.96 0 0 11.96 0 0 Tm 30.494 55.685 Td (tegers,) Tj )gp&(]I1OE%Od)LhrC3QU[O2W8 1.178 0 Td (S) Tj 1.713 0 Td (univ) Tj 0.438 0 Td (2) Tj 1.898 0 Td (7) Tj HX0JFV&@qB@jBl7R"AS-$,E-670A9Di60eb=)04SR-Ch[d"EbT0"FGQ++>72E@:NkhBkM+$+E2IF$=n9u+>GQ,+>7/@B4>mi@r#XtCh74#+E2IF$=n9 /F4 1 Tf 1.196 0 Td (b) Tj /F5 1 Tf /F14 1 Tf 1.087 0 Td (jects,) Tj qgOh%K9fTX)8Li!UC3g%J\b@2-:]`a/n[QF_U[62,c/"/Z?/FLrUQsLO,Rq$kpHSV\&FUa#6d"P"*= a;Sa.MAmIUdMHcmrH-Ghe$:4`42Ee$KR^ALZLs?! Tj 1.773 0 Td (ec-) Tj Ab#g6Dn#-[U=^&4j.Q9nDUY,53e2C7Ib;io5;Wrq^]DFduZ0,5OE0sd]?nAMkAqI(K4c+[#8Ns\kk8 3.805 0 Td (2.7.) 11.96 0 0 11.96 0 0 Tm 15.609 53.352 Td (\() Tj NEeN0%gMu-+d.0DuP`Srp!5$p]]:VkqOEU'>6l#X0hY7';).qT1[8iIoJ5=X9f0`JeZe`XrFil##d& -13.858 -2.582 Td (2.1.1.) 11.96 0 0 11.96 0 0 Tm 1.485 0 Td (of) Tj endobj 1.750 0 Td (same) Tj /F1 1 Tf NEeN0%gMu-+d.0DuP`Srp!5$p]]:VkqOEU'>6l#X0hY7';).qT1[8iIoJ5=X9f0`JeZe`XrFil##d& 1.060 0 Td (erations) Tj 2.713 0 Td (one) Tj 1.196 0 Td (w) Tj 4.324 0 Td (of) Tj endobj /FontDescriptor 24 0 R :bZW@NYjVS^V9c,Si!X__7jH-O)` 39 0 obj mW^S"S;#WPqO$6`F=1rDU^X/A@#"0C9j_)E0D 52.429 38.439 Td (2) Tj 11.96 0 0 11.96 0 0 Tm 25.259 22.176 Td (ositiv) Tj 0.834 0 Td (2) Tj /F5 1 Tf /F4 1 Tf ^MBDOGn8pq;%S4.3oE!VZPI0I[)AaAA]gpo.I:obHnXROdVq\0EK7ZC$d`4TCPu2tBH {a, b, c, …, x, y, z} is a set of alphabet letters. jVWdGL\[W\#AluEM'U=Wc2rAa_o!-RjM.A9o]q/aL#A 0JG170JG170JG170FBKf0JG170JG170JG170JG170JG170JG170JG170JG170JG170JG170JG170JG (PRS>OD*f-!#XhiM+,O0PRaXlR&4IH&/daH*Lt7F,\U(9T9s_r endobj 0.969 0 Td (w) Tj XS@A7]? /F5 1 Tf 0.840 0 Td (o) Tj 3.070 0 Td (Ph) Tj Resource added for the Mathematics 108041 courses. 11.96 0 0 11.96 0 0 Tm 1.748 0 Td (iden) Tj /Ascend 705 ]FAI6lEX\?%%4UTf(iNq<1jpEJ=%gkF@TU Tj /F14 1 Tf 1.738 0 Td (n) Tj 11.96 0 0 11.96 0 0 Tm 31.201 38.361 Td (\275) Tj 1.583 0 Td (alternativ) Tj [rOT\N*Im&lYSbn?5OmtHO_Bj$h.gT9@='LhZ)tj[N2`aX%7qNV*.JW30>C+8 A domain is a set of all input or first values of a function. 11.96 0 0 11.96 0 0 Tm 2.2 Ordered Pairs, Cartesian Products, Relations, Functions, Partial Functions Given two sets, A and B,oneofthebasicconstructions of set theory is the formation of an ordered pair, a,b, where a ∈ A and b ∈ B. kp3$9knBcqA;F:)Q$E$-kp3?TtqBQ##4F`[t$F`8HX3A;R/F(o9(@3BN3F:)Q$E$-kq1E\>iBPAT.F 0.00 g 0.00 G 0.500 0 Td (1) Tj c$Fld'\pQ"X\Lm(8qg)G\e#V;Orh;9S8@l_eJ\Fo!.8hBW9n^D\HaGntG'J0UJl^b-XK Let’s now review some key concepts as used in functions and relations. 4.285 0 Td (L) Tj 'b^c(9fS8D6L*c$?YUN=;4BAJopT&,AJj`$UD4g%CG2%@JALe8g,/&W%8f=odui`0K.qFroT&?jGAM 27.958 56.487 Td (2) Tj Example People and their heights, i.e. 0.944 0 Td (A) Tj IUirn+N=MD@39'*Y[=Z9oQdLHn-GM'8Xqe(J;g/@eOOq"SgdtT=@!.6?U<4SpI+-F+08_4(V^XsB-H;eLHE6 << Example 2: Find domain, co-domain and range of a Relation pRq = p divides q. 1.631 0 Td (taking) Tj 0.405 0 Td (\241) Tj Here 4 is not included in the set but 5 is included as x. 0.500 0 Td (\\) Tj /F1 1 Tf 11.96 0 0 11.96 0 0 Tm 37.887 53.352 Td (taking) Tj u+>GQ-+>7/@B4>mi@r#XtEbT0"FVR9DbcX)cHaH\/G#QkREXM0feq-IJNF0S`je.g:F!EI%C@4FG-!JTA /F5 1 Tf If R be a relation from A to B, then B is the co-domain of R. Range of relation R is the set of all those elements b ∈ B such that (a, b) ∈ R for some a ∈ A. /F1 1 Tf 0.500 0 Td (x) Tj -21.721 -1.574 Td (3.) Q q Found insideAs a result, this book will be fun reading for anyone with an interest in mathematics. +3Xpn]ZfEnN1(0?&>JFUsSdBp1Q2-XE%]L2_@YE9l-kp)U+^R$:NM&EHNPL3]H;a]75C0c93SYUMIW If C is a circle, then C ∈ Q. /F5 1 Tf 21.786 43.473 Td (,) Tj 0.444 0 Td (\242) Tj /Name /F5 /F4 1 Tf /StemH 20 /F15 1 Tf )gp&(]I1OE%Od)LhrC3QU[O2W8 \j3bC;";YJI6MUARrfGZ8)dCoC>TP>]4%*`pQ/n,tICO:79.IZgd!7d-]L,Bcr>OI[N ;2F`[t$F`8HX2`Mp5AT 2.189 0 Td (ha) Tj 0.889 0 Td (C) Tj /F4 1 Tf LoU,_NUR9E2"D(:?60^&39iU]\'L^XfBC'>='L.3J56j)GOMUlYOI"]g#_l%96Q@q7-dmAY"D)^jW^ -6m=(Gq! ET If P = {1, 2}, form the set P × P × P. Solution: We can simply write its 3 different element in a ordered triplet /Outlines 2 0 R 3.805 0 Td (2.1.) [61S=P8EdCV`QEl5I1k2)Q*soW+isId"^r-;"D9V\R3;2k`AT?r3kElQ12S==FbJ*bECmYUBplu6Pg 24.193 7.870 Td (A) Tj 0.381 0 Td (B) Tj /F5 1 Tf 1.528 0 Td (of) Tj iB8!uDEb]Y4ONN9?RdfE3EJZgR=McZ"-oJ>GAd3c$ka/KYNV0PLVeqnTSI0uR)_1LPUU^9g)J^317; /F4 1 Tf /F3 1 Tf [01KAeBl&&i@;TQu-pqoiE-686EZe(pA7]e!.3NYB@:X Answer (1 of 2): Relations are good for understanding how variables behavior together. 19.338 38.426 Td (Figure) Tj /F5 1 Tf 42.198 49.661 Td (2) Tj /F15 1 Tf XS/Y2frfRp0edRoI7#]SWrD6KJj`:jAlFoB1B!N+YS5csYrpb?H,OUP"I^"@U`G,E\jfO(:*5LDFQD 0.789 0 Td (ust) Tj 11.96 0 0 11.96 0 0 Tm 33.274 16.884 Td (could) Tj 3.682 0 Td (Y) Tj -22.740 -1.166 Td (3.) &.m0O5fAART+fDJXS@A7]? 0.278 0 Td (P) Tj 2.230 0 Td (j) Tj 51.751 47.724 Td (n) Tj 1.558 0 Td (theory) Tj 11.96 0 0 11.96 0 0 Tm 38.350 45.023 Td (other) Tj -28.463 -1.166 Td (builder) Tj /F5 1 Tf Q q P4iT*! "+KS9V$:$eS_9. 0.524 0 Td (:) Tj /F1 1 Tf p,r6SLrE3OB*Pfj:%1=QDmhJE!EWS1B#H.$-f!YJnP``YtL-`U%]Ch&n%f+oS!fEa"\r`nn$[jna#C A k-ary relation on Xis a subset RˆXk. /Length1 1922 ;2 2.066 0 Td (is) Tj 0.978 0 Td (gg) Tj 0.952 0 Td (b) Tj 11.96 0 0 11.96 0 0 Tm 17.023 49.604 Td (a;) Tj 1.183 0 Td (elemen) Tj 0.835 0 Td (and) Tj /Type /Font 0.555 0 Td (9) Tj 0.889 0 Td (\() Tj laRCC/1QhG`On@Pr@fH1&a\YBYZ3"@",R]22AX&sfF$0]"4dr_EmRPA@S]A\orI+j"^A`"OeJE]QNYb+'5Q6 0.975 0 Td (A) Tj A familiar example is the equality of two numbers. /F5 1 Tf /F4 1 Tf 57.079 31.782 Td (n) Tj And The Range is the set of values that actually do come out. /F4 1 Tf *d&e"XbV+bH=MFYL(lk*L'\QtQNkT[>Sq#4AOf./Cn^l9L6rY;^If _.R&nqk&G@MD*g? /F1 1 Tf 1.011 0 Td (=) Tj 1.728 0 Td (corresp) Tj 1.039 0 Td (A) Tj ET 11.96 0 0 11.96 0 0 Tm 24.258 27.518 Td (and) Tj a relation is any set of numbers that are able to be graphed on a coordinate (x, y) plane. Example 1: Is the relation expressed in the mapping diagram a function? 0.571 0 Td (e) Tj 3UY%^XmW@Yr]=cbd%**V(#P*cf_>`1h`)"3HHFeXR/J8@7*%[m6SEHCtD;.o;&'tL"BOiH4&UP(Y3e /Length1 2008 @(DEb/ZiDf0B:+Co%q$84hc@rGmlDJ((I2)ZRcEcPT6$8Pf3+F>gcBl7HmGT^RB@qZuGc/+>7SGDKU"CE-670A9Di60fCd005tKCBl@NlASuU$Eb/c(E-670A9D /F4 1 Tf 1.963 0 Td (in) Tj 0.856 0 Td (sets,) Tj JG170JG170JG170JG170JG170JG170JG170JG170JG170JEhf0JG170JG170JG170JG170JG170JG1 1.346 0 Td (that) Tj 0.500 0 Td (;) Tj 4.214 0 Td (represen) Tj Q /F1 1 Tf /Length3 532 /F1 1 Tf /LastChar 255 stream /F5 1 Tf c92)dQR2$tWP6Z6phEbT0"F^/ho%6.3NYB@:X:oCj@.6AS)9&:2b5g@q?c26Z6phEbT0"FdART+fDJXS@A7]? 357.2905 310.0159 l 352.1747 337.3001 l 335.1221 349.237 l 314.6589 347.5317 l h f* A relation may have more than 1 output for any given input. /F1 1 Tf 7.97 0 0 7.97 0 0 Tm /StemH 20 /F1 1 Tf 0.701 0 Td (If) Tj RelationRelation In other words, for a binary relation R weIn other words, for a binary relation R we have Rhave R ⊆⊆ AA××B. t$7er7o&5-I:P37DiXp`R+.? 0.438 0 Td (3) Tj Mrm)6A"V)fuZJ`%9m/+Db24C6#F,J0;I&"U,U+Yb[T9mT;u*l:6La7TDhQVm@78+VS:LQLRNW$ >> Tj endobj %2pF.@YrCLLki+*u3J:3?HJ!Tn`mhB)YgU! /Filter /ASCII85Decode 1.686 0 Td (same) Tj ?c2=2KpCGQ/6F-)lLY&(/;=6NGJLPCVqEfuWWLn1PYsZ0'/>%^#L"l5@JI!X\3[*-*p>X CK-12 Foundation's Single Variable Calculus FlexBook introduces high school students to the topics covered in the Calculus AB course. Topics include: Limits, Derivatives, and Integration. @C9? 3.154 0 Td (t,) Tj 11.96 0 0 11.96 0 0 Tm 28.086 39.275 Td (Its) Tj ET (Q>fAnb?h]6B1WU]J >]]mpMaNe-8D_o2&Vk^YQ60#lfc#2I8&b?OeW`\kU>XJDrUI,.W+7ndaooNWH. 1.184 0 Td (all) Tj >> /F1 1 Tf 0.680 0 Td (e) Tj A function from set P to set Q is a rule that assigns to each element of set P, one and only one element of set Q. 3.077 0 Td (of) Tj A?^Int/V1. i";'ir7YC07$"(u_8>rt.F@-k;A%d+[5f"FirOI_Q#qkK8laU^]Gfn[XP$eGW%Ae oOT@^N"+dKeXkAG&e1Z7A[)gAa+(9OM?g(MVJKql+WNu8O2Hqo,fI9mnM;]Y\ehQIfX>e82QQo22+3*)_.SoF]V!2GS//Y0pI Relations; Functions; Types of Relations; Representation of Functions Q q /F4 1 Tf ET /F5 1 Tf 1.588 0 Td (can) Tj /BaseEncoding /MacRomanEncoding The collection of all subsets of a set is the power set of that set. 11.96 0 0 11.96 0 0 Tm /F5 1 Tf /Parent 3 0 R Two sets are said to be equal; they contain the same members. 7.97 0 0 7.97 0 0 Tm Tj E:SZo,G16u5? 0.489 0 Td (;) Tj 0.438 0 Td (2) Tj 0.489 0 Td (;) Tj 0.381 0 Td (M) Tj 4.948 0 Td (Di\256er) Tj /F14 1 Tf 0.658 0 Td (=) Tj 0.489 0 Td (g) Tj -16.982 -1.582 Td (A) Tj 22.877 59.516 Td (N) Tj 2.233 0 Td (Symmetric) Tj Q JG170JG170JG170JG170JG0f0JG170JG170JG170JG170JG170JG170JG170JG170JG170JG170JG1 0.571 0 Td (ers.) Inverse relation is seen when a set has elements which are inverse pairs of another set. 0.956 0 Td (\\) Tj 0.489 0 Td (g) Tj 28 0 obj 29.632 76.911 Td (2.1.) -OTW+7M=.G\sn3iNXH9=1$,SR)[#?tTRhD.LV%! 7.97 0 0 7.97 0 0 Tm /F1 1 Tf 37.208 74.660 Td (A) Tj /F1 1 Tf is not a function since 2 gets sent to more than one value. P`Ah? 0.658 0 Td (=) Tj 0.444 0 Td (\242) Tj n]dTr)B;Zkq'D8?:Ii0n3opdh+`F;QjnJB@Nn!*e\qOH4WU5[ag++l-neU6? J?KL"RS$27R2Wn2PP&*FM'1@-_5Y'1IgA[?H2"2me>*mC_%tYjXZsC$-c#83?oq3#Xl&.d8c[I)'I81qCP/K 0.489 0 Td (;) Tj 0.734 0 Td (j) Tj 0.645 0 Td (=1) Tj 0.400 0 Td (qual) Tj /F4 1 Tf 0.923 0 Td (10) Tj 0.772 0 Td (j) Tj O)A+pE82Mq?g*s?3Y(2[Er? 1.039 0 Td (U) Tj 1.192 0 Td (\265) Tj 0.564 0 Td (aws:) Tj Tj 1.132 0 Td (of) Tj ;5`#:]cui:E.5r;+*tqUCXOV8DrdLE,ijB1UWZNGIpQ/%1X]I:>o;? J=H+M4H,eS8>GO8VVP-^9?_bNGnbjWU>\i#6=.G5BG,5QdRQ/Cp9`u /F4 1 Tf 11.96 0 0 11.96 0 0 Tm 35.688 24.495 Td (;) Tj /Type /Page 0.588 0 Td (their) Tj 1.412 0 Td (for) Tj /Pages 3 0 R BT /F5 1 Tf 0.956 0 Td (\241) Tj 1.903 0 Td (in) Tj 1.071 0 Td (in) Tj 11.96 0 0 11.96 0 0 Tm 25.351 52.186 Td (understo) Tj 0.603 0 Td (^) Tj 0.489 0 Td (;) Tj 6oh>NMC)7D@oiXHjWQ1ieY"6tF^]9Gp!#I"<8`1J*B=V:Dpu2*`$;* /F5 1 Tf /F5 1 Tf 0.438 0 Td (5) Tj /ItalicAngle -14 LcK0s2bhZAg]i(et-T'cGlfLBBM?,Db"qM\4A/2Gl=\4Rp13^nIoL\:.ohFji^oRr3nE_r(I1FW9I$ 1.491 0 Td (\241) Tj !6J"(oTA/7(1S=egoI3[aWBb,DncC] @'&ppeYEnDH?2>I9rj04jTJfR` The following are characteristic features of a function defined from a set X to a set Y: 11.96 0 0 11.96 0 0 Tm 27.775 20.817 Td (j) Tj endstream /F1 1 Tf /F1 1 Tf :95Kq1?j>lO!d*%D?AdV(NqIpem#E;5uQ?-+l"E(&XT]iO`"QU2Aui1 20.923 52.937 Td (A) Tj /F4 1 Tf 0.603 0 Td ([) Tj 0.889 0 Td (B) Tj /F1 1 Tf /F5 1 Tf Therefore, relation #2 does not satisfy the definition of a mathematical function. 0.500 0 Td (1) Tj 5.323 0 Td (L) Tj 1.093 0 Td (in) Tj )F-T-$jFIE50S*qZHf0@:#^6)2D.l Each element in a set is unique. 1.072 0 Td (=) Tj 1.691 0 Td (b) Tj << << "DE.FA9T``XAM_:YZEbQ0,8=f$G,9H+Uda^#CRJ0A$npb!delo3+PMY_oY;QL`QffulO?BaC<0N;lS 11.96 0 0 11.96 0 0 Tm 31.383 26.351 Td (j) Tj Jn&EHBE)Lq@B*5bPjHVNV$d'P'3nQAfeX*d4f\((-4/"#bh(nSu8s(Sk5KuUj_l!Dn^O5_^JnN0_%=)[0B>'fuE?DYk@gNgT8Ar][R97L-:[IE7mLs:L2bYROt5jXEWQlGTP@!&hN6 All the first values in W = {(1, 2), (2, 3), (3, 4), (4, 5)} are not repeated, therefore, this is a function. /F1 1 Tf 1.038 0 Td (b) Tj /F4 1 Tf >> /F4 1 Tf /F1 1 Tf 3.660 0 Td (Ph) Tj O)A+pE82Mq?g*s?3Y(2[Er? /F5 1 Tf /F4 1 Tf oWR``ca?Jt-ER. (qOrWn&,JcnWdhY5H],ZMi3h=AYqR, kT[>Sq#4AOf./Cn^l9L6rY;^If 0.888 0 Td (\() Tj 6kmk@&dl?j%8dIH 0.667 0 Td (;) Tj The arrow diagram shows one example of a one-to-many relation. nn+AN=kc:W-\k3RoOuZ*I(sb]/FO 0.1705 w Range of y = f(x) is a collection of all outputs f(x) corresponding to each real number in the domain. 0.803 0 Td (partition:) Tj 1.686 0 Td (range) Tj A “function” is a well-behaved relation, that is, given a starting point we know exactly where to go. 2.909 0 Td (ts) Tj 1.136 0 Td (Assume) Tj 5N7. /Parent 3 0 R 7.97 0 0 7.97 0 0 Tm /Descent 0 /F4 1 Tf /F5 1 Tf nca2ne:O'lhYlM%Sp+k2.7i,3eRUOK/Of*.OeGXK6XF(W>0ee*G(4gVjVKj@lKQ%Y0%AOm:[5O-9p`k2(*"-^l5 0.999 0 Td (B) Tj 1.116 0 Td (a) Tj Range : Elements of B involved in mapping. rnAtO3piA,rYsen4CXhDSmti7SO0VJ+$C*op)ub(`G$UE[IFVP\U;i&@CL;N? ? 0.999 0 Td (.) Q q Q 3.340 0 Td (in) Tj npNM=(e&cntS33 /F5 1 Tf /F4 1 Tf 11.96 0 0 11.96 0 0 Tm /F3 1 Tf /F4 1 Tf 11.96 0 0 11.96 0 0 Tm 11.96 0 0 11.96 0 0 Tm 26.533 35.792 Td (set) Tj Q q /F5 1 Tf In simple terms, I would define a function as a machine that takes an input, and returns an output by applying a specific rule to the input. Let sa... 0.381 0 Td (g) Tj /F1 1 Tf 0.794 0 Td (:) Tj .>ET+1/-;cL8&E@ZihJOk7\=S6Zu4bdi*GQr,SnH,L;_6h_@1lCXY!+Ou$=QV'=s]Z8EOZ`UW=S><6p]ZRcu:U?6,sY+,,0(kM*8@e,P/#ApUh_$>OlN(Hpja\R BT /F4 1 Tf /Widths [ 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 272 490 326 326 816 326 272 381 381 326 762 272 326 272 490 490 490 490 490 490 490 490 490 490 490 272 272 326 762 326 462 762 734 693 707 748 666 639 768 734 353 503 761 612 897 734 762 666 762 721 544 707 734 734 1006 734 326 326 272 490 272 326 326 272 490 544 435 544 435 299 490 544 272 299 517 272 816 544 490 544 517 381 386 381 544 517 707 517 517 435 490 979 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 612 816 762 680 326 734 707 762 707 762 326 326 707 571 544 544 816 816 326 299 490 490 490 490 326 326 435 490 707 762 326 326 993 762 272 490 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 326 ] 170JG170JG170JG170JEhf0JG170JG170JG170JG170JG170JG170JG170JG170JG170JG170JG170 &M?g(``Q6+?=L?nX;SlfpYJH%i!njC"3VI`1pJTQYcZ5>!]tNWJs^2Hfl? 0.489 0 Td (;) Tj 0.500 0 Td (.) 0.889 0 Td (\() Tj c$Fld'\pQ"X\Lm(8qg)G\e#V;Orh;9S8@l_eJ\Fo!.8hBW9n^D\HaGntG'J0UJl^b-XK /F5 1 Tf W7*E"f.=]/8O%UAD$4TqB3>WZ[FW@;)ZPYf ;0R[tQ@:X:oCj@.6AS)9&ki,0K2!K+Co%q$=e4!Eb0-1A8,Oq+D#G$$=e4!Eb0-1Anc'm+D#,/ARc(:S/#`=?S!3JojIF-e O'Dn`FF$=.s/\D\KY:%iPirr0ts/fn*BB(1MYS>%N#oVYBdGY^i?P$ 0.889 0 Td (B) Tj /F4 1 Tf /F4 1 Tf @C^j+G?W3WWnZh,=IWE.0M/"=RM6E+Nsg+^[>J /Length 100 0 R 1.196 0 Td (fg) Tj :a_-A&4,FbA#0+ /F1 1 Tf WL@f4]!C$?K[HnsKiTAh4+FpHNS0ECWB*3Ef5e#%]L82VuPkK8]d4Ej'Aq%0aH1dWJej2QhaQMZ`U3 11.96 0 0 11.96 0 0 Tm 31.475 41.526 Td (of) Tj 22.508 38.411 Td ([) Tj 2.714 0 Td (2.1,) Tj 11.96 0 0 11.96 0 0 Tm 15.016 40.441 Td (3) Tj /F12 1 Tf Qf6/?f?;Q?A;"0A)Xg*?ZG_QWTG>DV$c^&XAG#/U<90\Bf;GiT^+^DpH.)SP(Nu1:VZ:.SgV()K! << 1.093 0 Td (in) Tj 0.999 0 Td (B) Tj /F13 1 Tf 0.846 0 Td (prop) Tj 16.617 43.473 Td (;) Tj Solved examples with detailed answer description, explanation are given and it would be easy to understand This is the Aptitude Questions & Answers section on & Sets, Relations and Functions& with explanation for various interview, competitive examination and entrance test. 0.489 0 Td (;) Tj "m+>7hMEb/f)E-670A9Di62)6:-AoDg4+E2I 301.44 296.64 m 310.224 296.64 l 310.224 297.126 l 301.44 297.126 l f In The Mathematics of Love, Dr. Hannah Fry takes the reader on a fascinating journey through the patterns that define our love lives, applying mathematical formulas to the most common yet complex questions pertaining to love: What’s the ... /F4 1 Tf 0.829 0 Td (then) Tj 0.944 0 Td (A) Tj 3.186 0 Td (Ordered) Tj 0.571 0 Td (er) Tj /F4 1 Tf )5%023R.F`[t$F` "@V^1B(B A relation is any set of ordered-pair numbers. 11.96 0 0 11.96 0 0 Tm 31.897 13.296 Td (g) Tj /F1 1 Tf 2.391 0 Td (in) Tj << 0.517 0 Td (y) Tj 11.96 0 0 11.96 0 0 Tm 26.743 46.190 Td (t) Tj /F4 1 Tf There are 5 objects in set A and 3 elements contained by set B. 0.489 0 Td (;) Tj /FontDescriptor 40 0 R A#WaG[Qb:>)+OrQ5/h"e!3'Pa^B9PhcRYb!\Qb[g=t*e:tre's5s>@uk$$*M?SiIp711M[ao^".h&4 endobj The main topics of this course are (1) sets, functions, relations, (2) enumerative combinatorics, (3) graph theory, (4) network flow and matchings. 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