[PDF] Mathématiques discrètes 1ère année

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e ????? ???f?????X????Y? ??? ???f(x0) = 0? ???f(x0) = 0? ?????f= 0? ??? ??? ???? ????n?? ?f(n0)f(n)? ?8 >0;9 >0;8x??jxx0j< ?????jf(x)f(x0)j< ????V2 T??? ???f(x)2V? ?? ?f1(V)2 T? ????? ???x(f+R) =x(f) +R:x0(f) +R:(R)???? ????R??? ???f+R2h? x p=ab?????a= 1??b= 1?? ?? ??n??? ?? ?????? ??? ???n= 2k???? ?? ??????k?????n??? ????? ??? ?? ?????? ???? ???????q??r???? ???a=bq+r??0r < b? ??? ??x??? ?? ???? ??????? ??? ??? ?????x=nx0??n??? ?? ?????? ??x0?? ???? ??????? ??? ???x0<1? ???? ???? >0? ?? ?????? >0??? ??? ???? ????x2I? ??jxx0j< ?????jf(x)f(x0)j< ?

????? ?? ????? ???? ?? ??? ????? ?????? ???? ?????? ??????? ??? ???? ?? ???????n? ? ???? ?? ???? ?? ???????

djn d6=nd? i2Ix i X 1inx i=nX i=1x i ? ??? ????? ????? ???? ?? ?????? ??? ????0? Pn n X i=02 i= 20+ 21+:::2n Pn k=02k=Pn i=02i=Pn =02? ??Pk i=1ik? Q 1knk n=11? i=1xi=Pn j=0xj+1? S n=n(n+ 1)Sn? X 0in 0jp2 i+j=nX i=0p X j=02 i+j nX i=02 ipX j=02 j nX i=02 i(2p+11) = (2 p+11)nX i=12 i = (2 p+11)(2n+11) i=0aixi??g(x) =Pp f(x)g(x)? ?? ??????? ?????p? ??9k2N; n= 2k? ???8n;p2N;??9k;l2N??? ???n= 2k??p= 2l?????9m2N??? ???n+p= 2m? ????8a;b;c2C;9z2C??? ???az2+bz+c= 0? ??8x;y2N?(x;y)(x;y)? ???8x;y;x0;y02N? ??(x;y)(x0;y0)??(x0;y0)(x;y)?????x=x0??y=y0? ????8x;y;x0;y0;x00;y002N? ??(x;y)(x0;y0)??(x0;y0)(x00;y00)?????(x;y)(x00;y00)? ???8x;y;x0;y0? ????(x;y)(x0;y0)????(x0;y0)(x;y)? a6= 1; b6= 1??n=ab?? pgcd(a;b)??a^b?? ???? ????? ??????d??? ???d??????a??d??????b? ? ??????? ????? ?????x2+y20?? ?? ???? ? ???? ???? ??????? ?????x??y?x2+y20?? ??a??b???? ???? ??????? ??d??? ?? ???? ??a??b????? ?? ?????? ???? ???????u??v???? ??? ua+vb=d? ?d??? ?? ???? ??a??b?

8a;b2N;9u;v2N; ua+vb= pgcd(a;b)

????? ?a??b???? ?????? ??d???? ????? ?? ?????? ???? ???????u??v???? ???ua+vb=d? ???? ??????? ???? ?? ?????a??b??d???? ????? ?? ?????? ???? ???????u??v???? ???ua+vb=d? ? ??????? ?? ?????a??b? ?? ?????? ??? ???????u??v???? ???ua+vb=d??d??? ?? ???? ??a??b? ?? d??? ?? ???? ??? ???????a??b????? ?? ??????u??v???? ???ua+vb=d? ?? ???? ?? ?d???? ?? ???? ??a??b???? ????? ?????? ???? ???????a??b???? ???ua+vb=d? ?? ?????? ???? ?? ?????u??v???? ???ua+vb=d???? ???? ???????a??b? ??d??? ?? ???? ??a?? ??b? ???????a??b ? ??A?????B?? ????A??? ??????? ?? ?????? ?? ??f2H( (~f(x)) =f((x))?

8A??????? ?? ??????;8

?????? ??C;8x2A ;8f2H( ); (~f(x)) =f((x)) ?A??? ??? ??????? ?? ??????? ??? ?? ?????? ??C? ?x??? ?? ??????? ?? ?f??? ?? ??????? ??H( ??? ??? ??? ?????? ?????x2+y20?? ?? ??????? ?? ?????? ????? ?? ???? ?? ?? ?????x= 0??y=i ???????z= 0?

? ???? ?????? ???? ??? ??? ? ?? ??????? ?? ????? ? ???? ? ? ???? ???? ?????? ????x? ??x??? ??? ??? ?????

?? ?????? ?? ?????? ????y??? ???xy=yx= 1?? ?9!z2N;8n2N; n+z=z+n=n?? ?8x2R;??x6= 0?????9!y2R??? ???xy=yx= 1??

9z2N(8n2N; n+z=z+n=n??8z02N??8n2N; n+z0=z0+n=n?????z0=z)

8x2R;??x6= 0?????9y2R??? ???(xy=yx= 1??8y02R??xy0=y0x= 1?????y0=y)

A????B??B????A?

??B?? ??A?????B?? ????? ? ???? ? ??B?????A? ?? ? ??A?????B?? ??? ????? ???? ????n? ??????? ????? ? ???? ???? ??????? ???? ??? ???? ???? ??????n?? ?z+n=n+z=n??z0+n=n+z0=n? z=z+z0???z0??? ?????? ? ?????? =z0???z??? ?????? ? ?????? e cos(x+y) = cosxcosysinxsiny sin(x+y) = cosxsiny+ sinxcosy ?????x=y? a p2 a = (p2 p2 )p2 =p2 p2 p2 =p2 ???? ??? ????? ????n+ 1?

X??? ???8n2X; n0n?

???? ????? ??????? ??X?? ?? ? ??????? ????? ??? ?? ????? ???? ???? ???? ????p < n+ 1?? ? fn2N;1n3g=f3;1;2g? ???? ?fn2N; n334g? ?? ?fn2N; n??????6g?

X??x2Yg?

i=1Xi=;? fx; x2X??x2Yg? ????Y??? ??????? ????X\Y? ??????Tn i=0Xi?

XnY=fx2X; x62Yg?

?? ?????? ??? ??????? ??X??? ????P(X)? ?? ??????? ???(Y[Y0)c=Yc\Y0c?? ???(Y\Y0)c=Yc[Y0c? ?YY0? ?Y\Y0=Y? ?Y[Y0=Y0? ?Y\Y0c=;? ?Yc[Y0=X ???? ??????? ???YnY0=Y\Y0c? ?? A;B2P?A6=B?????A\B=;? ? ??????? x2X? ?? ?????? ??A2P??? ???x2A? ???? ?P=ffkp;k2Ng;fkp+ 1;k2Ng;:::;fkp+p1;k2Ngg?

0?P=fff:N!N;f(0) = 0g;ff:N!N;f(0) = 1g;ff:N!N;f(0) = 2g;:::g?

?????:R!R??? ????? ?? ??????x2???? ?????? ????x? ???? ????? ? ???? ??D=D0??A=A0?? ?? ???? ??????x2D??? ??????? ??x???f??f0???? ??????? ????? ? ???? f(x) =f0(x)? ???IdX(x) =x???? ????x2X? ??? ??????? ??????? ??f?? ??????? ?????imf? ?? ??????? ???f1(A) =D? ???? ??????? ??? ??y6=y02A?????f1(fyg)\f1(fy0g) =;? ???fjE(x) =f(x)???? ????x2E? ???gf(x) =g(f(x))? gg(x)?fg(x)?gf(x)? ??f(x) =x?g(x) =jxj? ???f(x) =pjxj?g(x) =x2? ????f(x) =x3?g(x) = 2x+ 1? ???f(x) = 1=(x2+ 1)?g(x) =x2+ 1? ?? ??????? ???fIdA= IdBf=f? ?????h(gf) = (hg)f? ??????? ??? ???? ????n?? ?ffn=fn+1? ?? ??????? ???fn=fn???? ????n? ??? ????f:N!N? ?? ??????? ???f?????? ? ???? ????n2N??n >0?????f(n)< n? ??????? ???f

A??f(X\Y) =f(X)\f(Y)?

?? ??????? ??? ???? ????n2N?????? ? ???? ???? ???? ?????? ??????? ??? ???? ?? ?????? ?? ?????? ??????

k??? ???2kn <2k+1? ?? ??? ??????? ???f?????? ? ???? ???(f1)1=f?? B ???? ????y2B?? ? ? ?? ???? ??x2A??? ???f(x) =y? ?? ??????? ??x02A?? ?? ??????g:B!A??? g(y) =??x??? ???f(x) =y?? ?? ? ?? ? ???x0?????? ?? ? ????g(f(x)) =x???? ????x2A? ?? ????g??? f 1:N!N x7!0f 2:N!N x7!xf 3:N!Z x7!xf 4:N!N x7!x+ 1f 5:Z!Z x7!x+ 1f 6:Z!Z x7! x f 7:Z!Z x7! jxjf 8:Z!N x7! jxjf 9:Z!Z x7!2xf

10:Q!Q

x7!2xf

11:Q!Q

x7!1=x f

12:Q!Q+

x7!x2f13:Q+!Q+ x7!x2f14:R!R+ x7!x2f15:R+!R+ x7!x2f16:R!R x7!x3 f

17:R!R

x7!sinxf

18:R![1;1]

x7!sinxf

19: [=2;=2]!R

x7!sinxf

20: [=2;=2]![1;1]

x7!sinx ??f:N!N?????? ??? ? ?f(n) =n=2??n??? ????? ?f(n) = (n1)=2??n??? ??????? ???f:Z!N?????? ??? ? ?f(n) = 2n??n0? ?f(n) =2n1??n <0? ????f:N!N?????? ??? ? ?f(n) =n+ 1??n??? ?? ?? ?????3k+ 1? ?f(n) =n1??n??? ?? ?? ?????3k+ 2? ???f:N2!N?????? ???f(n;p) = (n+p)(n+p+ 1)=2 +p? ??f:N!N?????? ??? ? ?f(0) = 0? ??n?? ???? ??? ????? ? ???? ???n=b0+b110 ++bk10k?

0(g) = 1g'? ???? ????? ?? ?0((f)) = 0( f'1) = 1 f'1'=f?? ?? ????

0??????

????[1;n+p]? ???(R2)2? f f ???n+ 1p? ???? ??X[1;n]?????X??? ???? ??? ??XY??Y??? ??? ?????X??? ???? ???? ??X??Y???? ???? ?????X[Y??X\Y???? ????? [1;p]? ????Y? ???????jXYj=jXj:jYj?

YX=jYjjXj?

???????jP(X)j= 2jXj? ? ?x2X????? (x) ='X(x)? ????? ??????? x62X????x2X[Y? ?? ???? ?????x2Y?? ?? ?????? (x) ='Y(x) +n jXjk:jXj=jXjk+1? ?? (f) = (f0)?????(f(x1);:::;f(xn)) = (f0(x1);:::;f0(xn))????f(xi) =f0(xi)????i= 1;:::;n? ????

P(X[Y)?

P(X)P(Y)?

2 '() =i;j???? ????2Sn+1? S ????Sn+1?????? ???(k) =0(k)????kn??(n+1) =n+1? ?? ? ????2Sn+1?? () =0? ???? jSn+1j=jSnj? (n+ 1)! = (n+ 1)jSn+1j= (n+ 1)jSnj= (n+ 1):n:::1? A ??n < p?????Apn= 0? ??np?????Apn=n!=(np)!? A n+1=Ap n+1=jA1j++jAn+1j= (n+1)jAn+1j? ????? ???? A p ???p? jP p(X)j=jPp(X0)j? p? ???? ?? ??? ??p= 0?? ???? ???n ?? ??? ??p=n?? ???? ???n n p =n np n+ 1 p+ 1 =n p +n p+ 1 n p =n!p!(np)! ???x? p? p+1? p+1=jPp+1(X)j=jPj+jP0j=n p+n p+1? p? ?? ?? ??? ?????? ?? ?????? ??????n1? ???????p1? ? ? 1 1 1 1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

(x+y)n=xn+n 1 x n1y+n 2 x n2y2++n n1 xy n1+yn nX k=0 n k x nkyk k=0(k nxnkyk=0

0x0y0= 1?

(x+y)n+1= (x+y)(x+y)n = (x+y)nX k=0 n k! x nX k=0 n k! x n+1kyk+nX k=0 n k! x nkyk+1 =xn+1+nX k=1 n k! x n+1kyk+n1X k=0 n k! x nkyk+1+yn+1 =xn+1+nX k=1 n k! x n+1kyk+nX k=1 n k1! x n+1kyk+yn+1 =xn+1+nX k=1 n k1! n k!! x n+1kyk+yn+1 n+ 1 0! x n+1+nX k=1 n+ 1 k! x n+1kyk+ n+ 1 n+ 1! y n+1 n+1X k=0 n+ 1 k! x n+1kyk n X p=0 n p = 2 n k=02kn k? k=02k13nk+1n k? k=0kn k n p =np n1 p1 ??p6= 0? n p+ 1 =npp+ 1 n p n+ 1 p+ 1 =nX k=p k p k=0 nk k? ??????? ??? ???? ???? ?????? ???????n?? ?Fn+Fn+1=Fn+2 n X k=0 n k 2 =2n n k2N?

P(N)????N?

?? ?????? ??????? ?? ????n2X???n62'(n)? x N=P ?xn< xn+1? ?fx0;:::;xng [Xn+1=X? ???? ?????N??? ???xNx? ???? ???? ????n < N?? ?xn< x? ????xN??? ?? ???? ????? ??????? ??XN? ???? ???? ????y2XN?? ?yxNx? ?? ?? ?fx0;:::;xN1g [XN=X? ?????x2X??x62 fx0;:::;xN1g? ?? ?? ?????? ???x2XN? ??? ?? ?????? ??? ???? ??????? X????N? ???? ????x2X?? ??????i='(x)? ?????i2[1;n]?? ???? ???f(i) =x?quotesdbs_dbs45.pdfusesText_45

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