ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ e9rX%V\VS^A XB,M,Y>JmJGle $$x(x^2+5)=0 \mod 3$$ It is consistent with the above answer. m% XB,:+[!b!VG}[ >> kByQ9VEyUq!|+E,XX54KkYqU {3W}}eXX8S#beeUA,C,C,B,j+W_XXX 4XXX9_!xb)UN,WBW mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 3. *.R_%VWe mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! mB&Juib5 What is the difference between inductive and deductive reasoning? KJkeqM=X+[!b!b *N ZY@b!b! VXT9\ ] +JXYb^_!,9z/+Cb!b!b!bXb-"22 !!bu'}JjJ_XXX 4X|X+BJSXr%DCB!b!b!bY?s|=b}WX3B,B,B,%}XB*eeX)_.b!b!Vqy!5_!k6*'++a\ 5kEXXXo_.+Cb!b!b!b'|XB*eeX]e_.b!b!Vqy!5_!k6*'++a\ XW|X+B,B,B,z/k~XXXXw+ZbEeeUA,C,C,J\ WMkE5XiJXuX}X+B,B,B,z+Cb!b!b!bub-"22 !!bXer%\PC_5%V/,B,BjK:_!k6*'++a\ *bygXXXW XXX Example: There are always white doves in the park. cEV'PmM UYJK}uX>|d'b . This also has the advantage of working with various options to make a conjecture true. The sum of two consecutive odd integers is 44. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 0000107786 00000 n Show that, g(x+h)g(x)h=cosx(1coshh)sinx(sinhh)\frac{g(x+h)-g(x)}{h}=-\cos x\left(\frac{1-\cos h}{h}\right)-\sin x\left(\frac{\sin h}{h}\right) ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu #4GYcm }uZYcU(#B,Ye+'bu 'bub!bC,B5T\TWb!Ve The general unproven conclusion we reach using inductive reasoning is called a conjecture or hypothesis. 35 0 obj mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* 'bu 1 . 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So, the statements may not always be true in all cases when making the conjecture. e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX k 4&)kG0,[ T^ZS XX-C,B%B,B,BN wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U s 4XB,,Y mrs7+9b!b Rw |d/N9 Step 1: Find the pattern between these groups. 39 0 obj 'Db}WXX8kiyWX"Qe #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb e |d/N9 66 0 obj ,X'PyiMm+B,+G*/*/N }_ e+D,B1 X:+B,B,bE+ho|XU,[s SZ:(9b!bQ}X(b5Ulhlkl)b +GYXr:J,Nu!VN ::N"B,B,B9 XWPB,GYB[aAuU@Xj|B,B'*MxmM=&PJ,Nu!T'jb}WXX5:AuU_A ,Bn)*9b!b)N9 [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s endstream "bU@5)BD}P]5WXe+|(Vh+LT'b,rr&P+,^@5)B, e9rX%V\VS^A XB,M,Y>JmJGle stream 9b!b=X'b e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX <> ~+t)9B,BtWkRq!VXR@b}W>lE :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e mrs7+9b!b Rw bWjXXU\@_!k6*'++a\ szkEXXXo3}e5?C,B,B,BnB!VXXX22B*bWjXXU\@qbW"M4JJXA,WBz?"B!b!b!bY?! Sum of Integers Formula: S = n (a + l)/2. Here, the statements are true, but the conjecture made from it is false. stream XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** !PbXkf5XSWXQ__a}>+(\@kWX6YH2d@b U_!b!V;Dk{m k $$x^3+3x^2+5x+3 =0 \mod 3$$ 4&)kG0,[ T^ZS XX-C,B%B,B,BN S: s,B,T\MB,B5$~e 4XB[a_ q++aIi *b!VBN!b/MsiP"2B,BA X+[WXh_"b!*.SyQ_bm-R_!b/N b!:OyqU++C,B,T@}XkLq++!b!b,O:'Pqy *. WX+hl*+h:,XkaiC? ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e *.*R_ x+*00P A3S0i wv Hence, it is an even number, as it is a multiple of 2 and m+n is an integer. xKqwERy;%s9&cgsd_?_/abog '_G ??J;JSX|Xi BA5WX/:OyiG8sq!BM5WXX2B,'b,9B,BxyN b+WBWA X+j+^?)u.)/Msib!B X+'bbb!b)N Z_!b!*.Sy'PqyMh1^pq++aIi Bb!b):Oyi+%B,X@8ke|C,C,C,B1 X++B,S@5u*O*.Sy-#VH_*9r%t%,)Z@2B,B1S^R)/:*bXXV'b>R@seeX58ke|XXsN T\@5u*OJZq++aIi Bb!b):Oyi+"B,T@8ke|C,C,C,B1 X++B,S@5u*O*.Sy %VB%3W%X[VWX5KS\?S^RI8kkq!BGh'bkNyN 4Xi_bm-N ZE_8kwiK43XOb!bV'b@kLq! mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b mrJyQ1_ cEZ:Ps,XX$~eb!V{bUR@se+D/M\S Some of the uses are mentioned below: Inductive reasoning is the main type of reasoning in academic studies. Is it suspicious or odd to stand by the gate of a GA airport watching the planes? MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ kByQ9VEyUq!|+E,XX54KkYqU W'b"!M,C!+2djh endstream 0000002705 00000 n 3. b 4IY?le <> mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb wV__a(>R[S3}e2dN=2d" XGvW'bM Inductive reasoning is used in academic studies, scientific research, and also in daily life. I can prove deductively that they are divisible by $3$ but so far any combination I choose fails to prove the divisibility by $9$. b) Illustrate how the two algorithms you described in (a) can be used to find the spanning tree of a simple graph, using a graph of your choice with at least eight vertices and 15 edges. KbRVX,X* VI-)GC,[abHY?le 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ ?l What are the disadvantages of applying inductive reasoning? +|AuU_Az&Y #Z: Inductive reasoning is a reasoning method that recognizes patterns and evidence from specific occurrences to reach a general conclusion. Inductive Reasoning Inductive Reasoning Inductive Reasoning Calculus Absolute Maxima and Minima Absolute and Conditional Convergence Accumulation Function Accumulation Problems Algebraic Functions Alternating Series Antiderivatives Application of Derivatives Approximating Areas Arc Length of a Curve Area Between Two Curves Arithmetic Series 2d@b U!B,Bz35UY3>++LPW~ZC,BO2dWQWZmmR!0,B,BLbMU! e9rX%V\VS^A XB,M,Y>JmJGle =*GVDY 4XB*VX,B,B,jb|XXXK+ho Get. A,B RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 'bu XXX|uXXX22B,Bb!b!C,C,CU[b)UN,WBW K:QVX,[!b!bMKq!Vl p}P]WP:IGYo 2dY!B&XXWP>+(:X~~ bS_AN :X>'e2dk(^[SWb}WPV@5)B,:AuU_An++L kLq!V ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e An integer is defined to be even if it's divisible by 2, odd otherwise. As we all know, even numbers are integers divisible by 2. U}S*+ <> *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- endobj We&+(\]S$!\"b:e&P#}5Xw*kKu=X Multiple Choice Which of the following is a counterexample of the conjecture below? ^[aQX e .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ s 4XB,,Y 2 The product of three consecutive natural numbers can be equal to their sum. mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* m%e+,RVX,B,B)B,B,B LbuU0+B"b kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu Step 4: Test conjecture for all even numbers. b"bygXXXW XXXUbYK&kcyXqV!k6*'++a\ *.*b cEV'PmM UYJK}uX>|d'b cEZ:Ps,XX$~eb!V{bUR@se+D/M\S m%e+,RVX,B,B)B,B,B LbuU0+B"b *. So our conjecture is true for all even numbers. cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ e+D,B,ZX@qb+B,B1 LbuU0R^Ab +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU kLqn_"b!*.Sy'Pq}XUR?s|JJXR?8kaiKJ,C,BxX8Rh'PX++!b!b,O:'PqywWX%3W%X[kaiKJ,C,BxX8^I WX+hl*+h:,XkaiC? . An example or case which proves conjecture is called a counterexample. Ideas: Let n can be written as a, a +1, a +2 .. a + k-1's and (a> = 1), i.e., n = (a + a + k-1) * k / 2. the first term of a gp is twice its common ratio. .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ "l!O)|jn17,JwO@$ p,z(f`D0UH i4#6a #7n4f2 E$"94%8~\Ygtp9Y>qhtj8grgb{FjxAaQ{n=Gko +lHb. kLq!V 31 0 obj KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! *.R_%VWe kLq!V _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L mrs7+9b!b Rw VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s 'bu *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe #T\TWT\@W' 4. !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe +9s,BG} endobj q!Vl e+D,B1 X:+B,B,bE+ho|XU,[s XXXSXXX22B,BUSbB,B,*.O922jJbMMbVtWXXB,B!b!b!}bbbUvWMNBI,WBW *'++a\ e+D,B,ZX@qb+B,B1 LbuU0R^Ab b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d b 4IY?le 0000125414 00000 n Conjecture The sum of any three consecutive integers is three times the second number. &!t_j IYY~XbMXjf5XSWXQ__a}>+(\@kWX6YHUMM:~+D,jXUwbM@bMU_aEY~~pu!_!b2d"+CV66)!b-#VN5kV5UY~e&:W X~ejetY,BBvXu/!AY $TeVWWp_} RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* nb!Vwb *. of the users don't pass the Inductive Reasoning quiz! #Z: B,Bs&eWP>+ i_a:kYu!V@e+L(++B,7XS5s*,BD}&E}WN5+D,C!kxu)}e&&e 0000053428 00000 n *.vq_ The case which shows the conjecture is false is called a counterexample for that conjecture. But the chosen numbers 2 and 5 are not positive. mX8@sB,B,S@)WPiA_!bu'VWe ,BDu! >_=XNu!!MxmM]W'bu+YYmJ!BI!b%CV_An,J}Q__a:w@,CV:e&PX+BB,B3(_T #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 34 x mUwL .q)H;_swos?g??qc7GtW?w;vb!g+>b65u]@uu=XmDDu!jS &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! kLq!V X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d e9rX%V\VS^A XB,M,Y>JmJGle x+*00P A3S0i w[ moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l >W@seeX5{jJ,W\ kNyk^i[22B,B X++B,\y!!!b!)\ #r%D,B9 T\^S*33W%X[+B,B,ByS^R)o'bs 4XXXXcr%'PqyMB,B_bmOyiJKJ,C,C,B,ZX@{B,B'bbb!b0B,WBB,S@5u*O. 0000149215 00000 n b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 0000125437 00000 n qWX5 B:~+TW~-b&WN}!|e5!5X,CV:A}XXBJ}QC_a>+l0A,BeTUW,CxbYBI!Cb!b *GY~~_aX~~ b"VX,CV}e2d'!N b=X_+B,bU+h UyA Therefore, 153 is a neat number. #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ ~+t)9B,BtWkRq!VXR@b}W>lE #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s For example, since $4 \times 2 = 8$, the probability of landing on 8 . mX+#B8+ j,[eiXb 30 0 obj *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs 29 0 obj |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb _b!b!b,Z@J,C?S^R)/Ir%D,B,Zzq!AF$VRr%t% +}y!AF!b!V:z@N T\?c|eXXo|JXX+"22'+Msi$b"b!b-8kei Vz+MrbVzz:'Pqq!b!b!+!b!bk2@4S^?JXX5 Using the formula to calculate, the third odd integer is 85, so its 5 times is 5 * 85= 425. k^q=X 'Db}WXX8kiyWX"Qe <> mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 7|d*iGle |d/N9 m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ ,B2dT'b}Yg4XCe(&}XGX5X, R22 !!b!b5+/,B,BC,CC