離散數(shù)學(xué)課后習(xí)題合集

Y8）X解PP。PQQPQR受騙了，則原句翻譯為(PQ)R。）(PQ)(PQR))解TPTQ)(TQR))=(FQR))F=Q=QQ=F當(dāng)Q=T時(shí)，上式=F；當(dāng)Q時(shí)，上式=T，因此公式存在成真解釋(P,Q,R)T,F)(P,Q,R)T,T)）(PQ)QR)P)解T當(dāng)P時(shí)TQ)QR)T)=1QQR)F)==QQR)當(dāng)Q=T時(shí)TTR)=R=F=F當(dāng)Q=F時(shí)F(FR)=R=TR==RT當(dāng)R時(shí)，上式=T(P,Q,R)T,F,T)(P,Q,R)T,T)）(PQ)QR)P)解T當(dāng)P時(shí)(TQ)QR)T)TQ)QR)QQR)時(shí)===T當(dāng)QTTR)R==TFRF時(shí)，上式=T，因此，公式存在成真解釋當(dāng)R時(shí)，上式=，當(dāng)(P,Q,R)T,T,F)(P,Q,R)T,T,T)）(PQ)QR)P)解T當(dāng)P時(shí)=(TQ)QR)T)2TQ)QR)F)=QR)(QR)時(shí)=Q=QT當(dāng)Q(TR)R=T=FR=TFRF時(shí)，上式=T，因此，公式存在成真解釋當(dāng)R時(shí)，上式=，當(dāng)(P,Q,R)T,T,F)(P,Q,R)T,T,T)）((PQ)R)QR)解TQ時(shí)=((PT)R)TR)=TR)TR=TFRF=T。當(dāng)R當(dāng)Q=F時(shí)((PF)R)(FR)==(PR)RT當(dāng)R時(shí)=(PT)TTT==FF當(dāng)R時(shí)(PF)F=PF==PT當(dāng)P=TPF=F，3(P,Q,R)T,F,F),(,T,F)因此，公式的成真解釋為；成假解釋為(P,Q,R)(F,F,F),(,T,T),(,F,T)。）(PQ)QR)P)解T當(dāng)P時(shí)TQ)QR)T)==TQ)TQTQ==TFQF=T。當(dāng)Q當(dāng)P=F時(shí)=(FQ)QR)F)TQR)=QR)=F=F(P,Q,R)T,F)因此，公式的成真解釋為；成假解釋為(P,Q,R)T,T),(F)。）(PQ)QR)P)解T當(dāng)P時(shí)=(TQ)QR)T)QR)F)QR)=Q=QT當(dāng)Q時(shí)TR)=T4TR)R==TFRF=T。當(dāng)R當(dāng)Q=F時(shí)(FR)=F=TF當(dāng)P時(shí)=(FQ)QR)F)(FQ)QR)T)=QR)=F=T(P,Q,R)T,T,F),(T,F),(F)因此，公式的成真解釋為；成假解釋為(P,Q,R)T,T,T)。）(PQ)Q(RP解T當(dāng)P時(shí)=(TQ)Q(RT=TQ)QR)QQR)=T當(dāng)Q時(shí)TTR)=T=F=FF當(dāng)Q時(shí)F(FR)=R=TR=TFRF=T。當(dāng)R=5F當(dāng)P時(shí)(FQ)Q(RF(FQ)QF)==QF)=T=QTTQF==F。當(dāng)Q(P,Q,R)T,F,F),(F,T)因此，公式的成真解釋為；成假解釋為(P,Q,R)T,T),(T,F,T),(F,F)。）(PQ)QR)P)）(PQ)QR)P)）(PQ)QR)P)）(PQ)QR)P)解）(PQ)((QR)P)(PQ)QR)P)=(PQ)QR)P)(PQ)((QR)P)(PQ)QR)P)(PQ)QR)P)(PQ)((QR)P)(PQ)(((QR)QRP)(PQ)(((QR)QRP)(PQ)((QR)P)6(PQ)((QR)P)(PQ)((QR)P){,QPQPPQ(PQ),。,,,,,,設(shè)集合是完備的，則由聯(lián)結(jié)詞集合的完備性定義知Pf(P,Q,R,)PQRP,Q,R,=F=TPf(P,Q,R,)P,Q,R,=F=Tf）(PQ)(PQ)解(PQ)((PQ)(PQ==(PQ)(PQ)(PQ)=(PQ)(PQ)=((PQ)P)((PQ)Q)7(PP)(PQ)(PQ)(QQ)==(PQ)(PQ)）(PQR(RQP解(P(QR(R(QP=(PQR)(PQR)==(PPQR)QPQR)(RPQR)QR=P）(PQ)QR)P)解(PQ)((QR)P)=Q(PQR)=P(PQ)(PQR)===((PQ)P)((PQ)Q)((PQ)R)(PQ)(PQR)）(PQ)(PQR解(PQ)(P(QR==(PQ)(PQR)(PQ)(PQ)(PQR)=====((PQ)(PQ(PQR)((PQ)P)((PQ)Q(PQR)((PP)(PQ)(PQ)QQ(PQR)((PQ)(PQ(PQR)8=((PQ)(PQR((PQ)(PQR(PQP)(PQQ)(PQR)(PQP)(PQQ)(PQR)==(PQ)(PQR)(PQ)(PQR)）(PR)(P(QR解=(PR)((P(QR(P(QR(PR)(PQR)(PQR(PR)(PQR)(PQ)(PR)((PR)QQ(PQR)((PQ)(RR((PR)QQ===(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)=====(PQR)(PQR)）(PQ)QR)P)解(PQ)((QR)P)==(PQ)((QR)P)((QR)P)(PQ)(PQ)(PR)(PQR)=9(PQQ)(RR(Q(PP)(RR((PQ)(RR((PR)QQ(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)==(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(===PQR=）(PQ)R解PQR===(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)(PQR)=(PQP）P解P(P(QP==(PP)(PQP)T=T=T10=()(=(PQ)(PQ)(PQ)(PQ)=）(PQ)(PQ),(PQ)QP)解）(PQ)(PQ)=(PQ)(PQ)=(PQ)(PQ)=(PQ)QP)=(PQ)(QP)=(PQQ)(PQP)F=F=F()=）(PR)QR),(PQ)R解(PR)QR)=(PR)(QR)=((PR)QQ((QR)(PP=(PQR)(PQR)(PQR)(PQR)=(PQR)(PQR)(PQR)=11(PQ)R=(PQ)R=(PQ)R=(PR)(QR)=((PR)QQ((QR)(PP=(PQR)(PQR)(PQR)(PQR)=(PQR)(PQR)(PQR)=12(PP)）PPP1(PR)QR)((PQ)R))））(PP)((PP)((PP)P))Q,RP用））(PP)((PP)P)）(PP)P）(PQ)(PP)P）PQ用P(PQ)QP)(PQ))）7(P(PP))(((PP)P)(P(PP)))QPP用））((PP)P)(P(PP))）(PP)P）(PQ)(QP)(PQ)QP)））(PQ)(QP)PP用QQ用PP））(PQ)((RP)(RQ))）(PP)((QP)(QP))P用PQ用PR用Q13(QP)(QP)）3(PQ)QR)(PR）((PQ)(QP(((QP)(QP((PQ)(QPP用PQQ用QPR用QP((QP)(QP((PQ)(QP）））(PQ)(QP)QQ）QQ)((PQ)(PQQRP用P用Q代入，Q用(PQ)(PQ)）））((PQ)(PQ(((PQ)(QP((PQ)(QPPQQPQRQPP用用用））((PQ)(QP((PQ)(QP）(PQ)(QP)）P）P(PQ)P(PP)QP用）P(PQ)(QP)(P(PP((PP)P)QPP用）(PP)P））14(PQ)）QP(PP)）QP用(PQ)QR)(PR3）((PP)P)((P(PP((PP)(PPP用(PP)Q用P代入，R用PP(P(PP((PP)(PP））(PP)(PP)）(PP)P）((PP)(PP(PP)）PPP用PP））((PQ)R)((PQ)R)PP）1）QQ用PQ）(PQ)((RP)(RQ）(QQ)((PQ)(PQ(PQ)(PQ)PQ用RP用）3(PQ)QR)(PR）((PQ)(PQ(((PQ)R)((PQ)RP用PQ代入，Q用PQ，R用R((PQ)R)((PQ)R)）:PQP)AB:QR)((PQ)(PRC:(PP)P15D:Q(PQ)E:(PQ)QP)P）P）(PP)(RR)(PP)P）C(PQ)(PP)）Q）PDQ用PQR)((PQ)(PRB((PP)P)((P(PP(PPQPP用）R用P(P(PP(PP)）））PP）(PP)((RR)(PPPPRR用Q用P(RR)(PP)）））(PQ)QP)E）((RR)(PP((PP)(RRP用RRQ用PP）(PP)(RR)）(PQ)((PQ)P)PQQP16）PPP）P）Q））Q）PQ，PQ├P(PQ)((PQ)P)）(PQR((PQ)(PRQR)QPP）PQR）QR））PQR)，PQ，P├R(PQR((PQ)(PR）((PQ)R(PQR(PQ)R）P）QQ(PQ(PQ)PQ）17QPR））(PQ)R)PQR，，├((PQ)R(PQR）((PQ)((PR)QS(SR)）(PQ)((PR)QS89(PQ)P））(PQ)Q）((PQ)((PR)QS(PQ)QQ(PR)QS)用P用P）((PQ)((PR)QS((PR)QSQQ(PR)QS)P用P用QP））(PR)QS)）PQ）））((PR)QS(PR)用PPRQQS，用）((PR)QSQS)用PPRQQS，用RSPQ））RS））Q(PQP18(S(RSP用RQ用S）R(RS)SSR））(PQ)((PR)QSRS├((PQ)((PR)(QS(SR)）((PQ)((PR)QS(SR)((PQ)((PR)QS(SR)((PQ)((PR)(QS(SR)=Q(PR)(QS)(SR)=P）P）Q）PR）QS）SR）R）S）Q）(PQ)((PQ)P)(PQ)((PQ)P)(PQ)((PQ)P)==(PQ)(PQ)P)19）PQ）PQ）P）Q）Q）(PQ)(QR)RP(PQ)(QR)RP=(PQ)(QR)RP）PQ）QRR））P）Q）Q）PP(PP)PP=）PP）20解(e,e)e1設(shè)Ae122B(e,e)ee；1212ab(a,b)B(a,b)。解(e,e,e)設(shè)Ae1ee2；1233B(e)eC(e)eD(e)eabc(a,,c)B(c)C(c)D(c)。解(e,e,e)設(shè)Ae1ee2與3123abc(a,b,c)。解(e,e)e1設(shè)Ae；122bca21(a,b)(a,c)。解(e)e設(shè)PM(e)e(e,e)e犯e。1212x(P(x)y(M(y)(x,y。解設(shè)O(e)eP(e)ex(O(x)P(x。解(e)設(shè)Pe(e)ex(P(x)(xx(P(x)(x。解設(shè)F(e)eB(e)eW(e,e)ee。1212ax(F(x)W(a,xx(B(x)W(a,x。22解(e)設(shè)PeC(e,e)e犯e。1212ax((P(x)C(x,aC(a,xx((P(x)C(x,aC(a,x。解設(shè)L(e)eM(e)eC(e,e)ee。1212x(L(x)y(M(y)C(x,y。解(e)設(shè)Pe(e)eB(e)eF(e)eC(e,e)e為e1212D(e,e)e為e1212y((P(x)(yC(x,yy((P(x)F(yD(x,y解(e)設(shè)Pe(e)e23B(e,e)ee；1212C(e,e,e)eee。123123y((P(x)(y)B(x,yz(P(z)C(z,x,y解(e)設(shè)AeB(e,e)e比e1212x((x)y((y)B(y,x解(e)設(shè)AeN(e)eW(e,e)ee。1212x((x)y(N(y)W(x,y解(e)設(shè)AeN(e)eW(e,e)ee。1212x((x)y(N(y)W(x,y解(e)設(shè)AeB(e,e)e與e121224y(((x)(yB(x,y(e)e令PE(e)eO(e)eD(e,e)ee1212）E(2)P(2)解22）x(D(2,x)E(x解2）x(E(x)D(2,x解2）x(E(x)y(D(x,y)E(y解）x(E(x)P(x解）x((x,y)y(B(x,y)C(z解(x,y)x和B(x,y)x受xB(x,y)y受y(x,y)xB(x,y)x和y(x,y)y和C(z)z25）x((x)B(x,ty((y)B(x,y解(x)B(x,t)x受(y)B(x,y)y受(x)B(x,t)x和(y)B(x,y)y(x)B(x,t)t和(y)B(x,y)x(x)y(B(x,y)P(y2xy3。解=(x)u(B(x,u)Pu2xy3xA(x)u(B(y32,u)P(u))(e,e)(e,e,x,y)。B1212解(x)u(B(x,u)Pu(e,e,v,y)12()vC(,,,)(x,y)在個(gè)體域解個(gè)體域26XXX1232TFTFFTTFFFTTTFFTTTTF122122X)2..16X)1。ii。x()(xxxrx(）r和解()(()((=xxxxxxxxr==()())xxx）解()(())()xxxxx(x))xxr())x(X(x)Y(x(x)xY(x解1域=(XY)(XY)T12域2eX1T2T1X2X3XF4FTTFF(I;X,Y)X,X)3下2x(X(x)X(x))(xX(x)xX(x))=232327(FF)F=T=T=F2122k(k2kk2kx(X(x)Y(x(x)xY(x解k域,k}x(X(x)Y(x(XY)(X(2)Y(2))(X(k)Y(k(XX(2)X(kYY(2)Y(k(x)xY(xk域,k}）x((x,y(zY(z)Z(x解=x(x,y)(zY(z)Z(x=x(X(x,y)(zY(z)Z(x=z(X(x,y)Y(z)Z(xz(X(x,f(xY(z)Z(xx(X(x,f(xY(g(xZ(x）z(X(x,y,z)(uYu,x)xW(y,x解=z(X(x,y,z)uYu,x)xW(y,x28=z(X(x,y,z)uYu,x)vW(y,v=zuv(X(x,y,z)(Yu,x)W(y,v((,,)((,)(,yzuvXayzYuaWyv((,,)((,)(,(,,yzuXayzYuaWyfxyz（解(e)令Pe(e,e)ee；1212a!x(P(x)(x,a解(e)令Pe(e,e)e比e1212B(e,e)e關(guān)e；1212bc(()(,)))PxzPzAxz(()tyxBt,b)Bt,c)。解(e)令Pe(e,e)e比e1212B(e,e)e比e1212(()(,)))PxzPzAxz(()tyx29x(P(x)Bt,x。解(e)令Pe(e,e)e是e1212x(P(x)!y(P(y)(y,x。30x(P(x))xPx())）x(P(x))xPx())x(P(x))Px())）x(P(x))xPx())）全）2xP(x))xP(x))xP(x)P(x)）xP(x)P(x))xP(x))P(x)))xP(x))P(x))）））xP(x))xP(x))）xP(x))xP(x)))xP(x))xP(x)))x(P(x))xPx())))7）xP(x))xP(x)))xP(x))xP(x)))））x(PxxP(x))())）x(P(x))(xP(x))）x(P(x))(xP(x))x(P(x))(P(x)))））x(P(x))(xP(x)))存）1(xP(x))x(P(x))31P(x)xP(x)3(P(x)xP(x(x(P(x))(P(x)x(P(x))(P(x))）））x(P(x))xPx(())））x(P(x))(xP(x)((x(P(x))x(P(x)x(P(x))(xP(x)7）(x(P(x))x(P(x)x(P(x))(xP(x)））x(P(x))(x())）()(PQP(B)(PP)((x(xx├1212()x(x)。├0()x）x(x))(）A())(x）））x)(x))(A()x))(）））xx))(））)B())xx(））32x(x)））y(xy)y(zy)②b③z④z(zb)全0b⑤b⑥x(xx)全0解0xP(x)xP(x)①xP(x)②P(e)③xP(x)全0解0x(P(x)Q(xxP(x)②③P(c)Q(c)④P(c)⑤Q(c)xQ(x)⑥解P(c)cx(P(x)Q(xxQ(x)xR(xxP(x)xR(x33x(P(x)Q(xxQ(x)xR(x)xP(x)P(x)Q(x)P(x)Q(x)xQ(x)全0xR(x)x(P(x)Q(xxQ(x)xR(x)xP(x)xR(x)，，├x(P(x)Q(xxQ(x)xR(xxP(x)xR(xxP(x)x((P(x)Q(xR(xxP(x)y(R(x)R(y，├xP(x)x((P(x)Q(xR(x）xP(x)x((P(x)Q(xR(xP(a)P(a)(P(a)Q(aP(a)Q(a)(P(a)Q(aR(a)R(a)Pb)Pb)(Pb)Qb34Pb)Qb)(Pb)QbRb)Rb)R(a)Rb)(R(a)Rby(R(a)R(yy(R(a)R(yy(R(a)R(yy(R(x)R(yy(R(x)R(yxP(x)x((P(x)Q(xR(xxP(x)y(R(x)R(y，├(e)令Pe(e)eB(e)eW(e,e)ee；1212C(e)eD(e)e）x((x)B(x）y((P(x)B(y)W(x,yD(x35）y(P(x)C(x)(y)W(x,yx(P(x)C(x)D(x(x)B(x)）11P(x)B(y)W(x,y)D(x)）222）P(a)）C(a)）b)(a,b)）WP(x)C(x)D(x)））C(a)D(a)）D(a)333{/}ax3）P(a)B(y)W(a,y)）B(y)W(a,y)）Bb)a/x}結(jié)2b/}）b)a/x}1x((x)B(x））y((P(x)B(y)W(x,yD(xy(P(x)C(x)(y)W(x,y）P(a)C(a)b)W(a,b)）(P(a)C(a)b)W(a,bP(a)36(P(a)C(a)b)W(a,bC(a)））(P(a)C(a)b)W(a,bb)(P(a)C(a)b)W(a,bW(a,b)）P(a)）C(a)b)））(a,b)W）b)Bb)Bb)））））））(P(a)Bb)W(a,bD(a)）P(a)Bb)W(a,b)D(a)P(a)C(a)D(a)(P(a)C(a)D(ax(P(x)C(x)D(x）x(P(x)C(x)D(x））x(P(x)Q(xxQ(x)R(xxR(x)xP(x)），，├(x)Q(x)）P11Q(x)R(x)）22(x)）R3）P(a)Q(a)a/x}137）R(a)a/x}2a/x}3）y((P(f(xQ(fb(P(f(aP(x)Q(yy((P(f(xQ(fb(P(f(aP(x)Q(y=y(((P(f(xQ(fb(P(f(aP(x)Q(y=y(P(f(xQ(fb(P(f(aP(x)Q(y(f(x）P1）Q(fb))P(f(aP(x)Q(y)）））2P(x)Q(y)a/x}21P(x){fb)/}2{f(x)/x}12(e)e令P(e)eB(e)eW(e,e)ee1212）x(P(x)y((y)W(x,y38）x(B(x)(x）x(P(x)y(B(y)W(x,yx((x)B(x(f(xP(x)）A）W）A11(x,f(xP(x)222(x)B(x)33）P(a)）B(y),W(a,y)(x)(x)）B）(y),W(a,y)44{y/x}4P(x),W(a,f(x{f(x)/}）111）W(a,f(aa/x}1）P(a)a/x}2是在在在是在在和）A和BB和A39(e,e)令A(yù)e為e1122B(e,e)e為e1212M(e,e)e和e1212C(e,e)e為e1212D(e,e)ee1212E(e)eP(e)eF(e)eG(e)e）(，PD)）B(Smith，PD)）B(，PD)）B(PD)）(Hall，SD)）B(SD)）B(Bell，SD)(Jones,Mary)）M）E(Mary)）E(Jones)）F(a)40）B(y,a)P(y(y,x)）P(c)）Fb)）(c,b)）C(c,z)(z,b)(x,y)M(y,x)）M1111）G(Jones)）G(Mary)(x,)B(x,SDE(x)）D222問(wèn)）B(Mary,SD)在）E(Mary)M(x,Mary)）3{Jones/x}3。）D(Mary,)）B(Mary,SDE(Mary)）E(Mary)/x}241）xy。解2()Nyx,c）a為b解N2(N2rs(a,b)N2rs(a,c))）x解N2(Nrs(x,t)2)tx）x為yx解x為yN2rs(x,y)xN2(x[x]2)N2(N2rs(x,y)N2(x[x]2))）xx解xNxxN2(x[x]2)NxN2(x[x]2)a，xAb(x)解N2(ctA(a)ctA(actA(b))42a，xAb(x)解(a)(ab)(x,x,x)f(xg(x,x),g(x))）h123211322解h(x,x,x)f(h,h,h,h)(x,x,x)1231234123h(x,x,x)I(x,x,x)11233,2123h(x,x,x)S(x,x,x)52123123h(x,x,x)g(I,I)(x,x,x)312313,3123h(x,x,x)g(I)(x,x,x)412323,2123(x,x,x)fx,g(x,x）h1244112解h(x,x,x)f(h,h,h)(x,x,x)124123124h(x,x,x)S(x,x,x)31124124h(x,x,x)I(x,x,x)21243,3124h(x,x,x)g(I,I)(x,x,x)312413,2123dv(x解3dv(x3的倍數(shù)不為3的倍數(shù)1dv(x3N3rs(xN2rs(x3Nrs(xNrs(x212Nrs(x43）D(x)D(0)0D(xxB(x,D(x))(x)B為IDD(x)）ax1a0aaBxa(,)ax1xx(I,S)B為axa2,22,1ax(x,x,,x,x,x)）Im,n12nm1m(x,x,,x,x,x)Im,n12nm1m(x,x,,x,x,x)Im,n12nm1mmax(x,y)）x,y)x(yx)xyxymax(x,y)max(x,y)和xy和yxy）xy(xy)(yx)xy和xyxyxyxy和xyx44f(x)）xnfx()(0)f0xf(x)f(nf(x)B(x,n,f(xxn1xntn(f(SI),I)f(x)為B3,23,3xnf(x)xn45集合解解}解}解}。（5）a,}a,,c,{a,,}}解a和b均是集合a,b,c,{a,,}}（6）a,}a,b,c,{a,b,}}解a,}a,b,c,{a,,}}中無(wú)元素a,}。（7）a,}a,b,c,{{a,}}}解a和b均是集合a,b,c,{{a,}}}（8）a,}a,b,c,{{a,}}}解a,}a,b,c,{{a,}}}a,}46a,a,}。設(shè)Aa,,{a,},}a,}a,},}a,b,{a,},}}a,b,{a,）A）A）A）a,}}A）A）}A,BCAB,BCAC且。A和解B}CAB,BCAC且。令A(yù)，，A、B和CABAC，BC。解B為CA為BACAB，BCAC。解}B},b,}C},b,c,d}令A(yù)，，。AB，BCAC。注意應(yīng)為},b,c,d}。C，BCAC。A解a,}Ba,b,}Ca,b,},d}令A(yù)，，。。C，BCAC。ABAC，BC。A解a,}Ba,b,}Ca,b,},d}令A(yù)，，B，BCAC。A47,N,T,PSNM設(shè)SMTPT：M。P：T。M：P。P：M。(MP)：TCBCAACBCBA。xCAxCxC或。xACxBCACBC以，xBx。C時(shí)，由補(bǔ)的定義知xC，由交集的定義知xAC，因?yàn)椋?）當(dāng)xACBCxBCxB。BA。AA(CC)(AC)(AC)，CBCACBC且AAA(CC)(AC)(AC)(BC)(BC)B(CC)BUBBUBB故A。設(shè)A、B、C48）(AB)CA(BC)x(AB)CxABxCxAB且AxB且xBxCxBCxA(BC)得x和得(AB)CA(BC)。xA(BC)xAxBCxBC且BxCxAxB和xABxABxC和得知，x且。由得，又由x(AB)CA(BC)(AB)C。(AB)CA(BC)。(AB)C(AB)CA(BC)A(BC)A(BC)）(AB)C(AC)B(AB)C(AB)C(AC)B)(AC)B(AC)B）(AB)C(AC)(BC)(AB)C(AB)C(AC)(BC)(AC)(BC)(AC)(BC)B,CD(AC)(BD)嗎？又是否（1）設(shè)A，那么一定有(AC)(BD)解(AC)(BD)xACxAxC或。AABxBxBD(AC)(BD)。若x49CCD以xDxBD(AC)(BD)。若x(AC)(BD)。(AC)(BD)xACxAxC且AB,CDxBxD且xBD(AC)(BD)。X且WY)(XZ)WY)(XZ)解WY)(XZ)a,b,c,d,}Wa,}Xa,b,}Yc,d}Zc,d,}。U，，，，Ya,b,}XZa,b,}WY)(XZ)，。從而有，WWY)(XZ)a,b,c,d,}Wa,}Xa,b,}Yc,d}Za,c,d,}。令全集U，，，，Ya,}XZ}WY)(XZ)，。從而有，W9BACBCA解a,b,}Ba,}C}ABACBC。令A(yù)，，BACBCA解a,b,}Ba,d}Ca,}ABACBC令A(yù)，，BACBCA解CB。BB(AA)BA(AB)50A(AC)(AA)CCC設(shè)C）(AB)(AC)A解(AB)(AC)AABC①ABCxABCxAxB且且xC(AB)(AC)AxA得x(AB)(AC)x若xAB若xAC或ABxAC。xAxBxB且xAxCxC且ABC。x(AB)(AC)xABxAC或。若xABxAxACxA(AB)(AC)A；或xAABCxBxC。若xB，則xAB(AB)(AC)；若xC，則xAC(AB)(AC)A(AB)(AC)。(AB)(AC)A。）(AB)(AC)解(AB)(AC)ABC①ABCxAxBC但xBxC或。若xBxAB(AB)(AC)(AB)(AC)x若xCxAC(AB)(AC)(AB)(AC)xABC。51假設(shè)(AB)(AC)，則x(AB)(AC)，根據(jù)并集的定義知，xAB或xAC。但若xABxAxBxBCxAxBC且ABC若xAC且xAxCxBC，即xAxBC但ABC(AB)(AC)）(AB)(AC)解(AB)(AC)ABC①ABCxAxBC但xBxC且xAxB且得xABxAxCxACx(AB)(AC)，且得(AB)(AC)xABC。②(AB)(AC)x(AB)(AC)xABxAxBxCxBxCxBC且xACABCx得但且AxBC且得(AB)(AC)。）(AB)(AC)解(AB)(AC)ABAC①ABACxABxACxACxAB但或但。ABxACxABxAxBxAxAC且若x但由得且C得x，由xAxC和xACxAxB且xAB得，因此得，由x(AC)(AB)(AB)(AC)(AB)(AC)x。ABAC。xACxAB但若②假設(shè)(AB)(AC)，則x(AB)(AC)，根據(jù)對(duì)稱差的定義知xAB且xAC或xAC且xAB。52xABxACxACxAxAxCxA若且得或且xABxABxAxCxACABAC。且xABxAxB且xABxAB(AB)(AC)故。xACxAB且若PQ和PQP））））PQPPQPPQPQ解PQPQ）））QPQ）ABAB和和ABBABBA））解ABAB））ABCU設(shè)、和是ABACABAC和BC解BCB(AA)B(AB)(AB)(AC)(AC)(AA)CCBCxxCxC但xB或xB但xCxAxA或xA（153xAB為ABAC以xACxCxC矛AxAxABABACxxACxCxCBC。C但xBxAxA或xA（2xxACABACxABxBxB矛AxAxACABACxxABx）}BxBBC。））,{解2,{}}））A2,{{}}}A2,{},{{}},{,{}}}）A}B，((A設(shè)A）B？）B？）}B？？）}B）}}B）B？？：設(shè)A解a,{}}}()54}()？}}()？()a,{}},?A}()}()？}}()？()？設(shè)S,T,M）是TSMTMSS。TST。TUS。SS）S解9nnn2。n2當(dāng)n時(shí)(n2)(nn012333=333k(2)(3k39m0k，當(dāng)n3k(k2)(kk9m333。nk1時(shí)55(kk(k(k2(kk333333(k2)3*(k2)*33*(k2)*33(kk===322333(k2)(kk9(k2)27(K2)2733329m9(k2)27(K2)2729m9(k2)27(K2)27為92，2n11122n1n0時(shí)，當(dāng)nn22n121k1112m021當(dāng)n2kk1112=133mk22k1。當(dāng)nk1時(shí)111211*1112*12k32(k1)1k22k1211*1111*12*122k1==22k1k11*12)133*12k22k12k111*133m*122k1=133*m12)=2k156設(shè)A()A。解(),()A{((,2),C,BDABCD。A(x,y)ABxAxB且。因?yàn)閷?duì)于，則根據(jù)笛卡爾積集的定義知AC,BDxA得xCxB得xD(x,y)CDABCD。ABCD,ACBD解BCDABCD，令A(yù)D，，，但B。AABAB，和AAA）AA解A。B。）A或A。設(shè),B,C,D(AB)CD)(AC)(BD)(AB)CD)(AC)(BD)57(x,y)(AB)CD)xAByCD，AxByCyDxAyC(x,y)ACxByD和知，x，，，和得(x,y)BD(x,y)(AC)(BD)得。由交集的定義知，故(AB)CD)(AC)(BD)。(AC)(BD)(AB)CD)(x,y)AC(x,y)BD(x,y)(AC)(BD)有且AyCxByDxAxB和得x，，，xAByC和yD得yCD(x,y)(AB)CD)(AC)(BD)(AB)CD)。(AB)CD)(AC)(BD)。(AB)CD)(AC)(BD)解BCD令A(yù)，，，(AB)CD)(2,4)}(AC)(BD)(2,4)}，。則(AB)CD)(AC)(BD)。(AB)CD)(AC)(BD)解BCD令A(yù)，，，(AB)CD)(AC)(BD)，。則(AB)CD)(AC)(BD)。(AB)CD)(AC)(BD)解58BCD令A(yù)，，，(AB)CD)(AC)(BD)，。則(AB)CD)(AC)(BD)設(shè),B,C(AB)C(AC)(BC)(AB)C(AC)(BC)(x,y)(AB)CxAByCxAB，AxBxAyC(x,y)ACxByC得x和。由和得；由和得此(x,y)BC(x,y)(AC)(BC)，從而有，因(AB)C(AC)(BC)。(AC)(BC)(AB)C(x,y)(AC)(BC)(x,y)AC(x,y)BC(x,y)AC且AyC且(x,y)BCxByCxAxBxAB和得，得x得且(x,y)(AB)C(AC)(BC)(AB)C。(AB)C(AC)(BC)。(AB)C(AC)(BC)(AB)C(AC)(BC)(AB)C(AC)(BC)解(AB)C(AC)(BC)．(AB)C(AC)(BC)59(x,y)(AB)CxAByCxAB，AxBxAyC(x,y)ACxByC和得x或。由和得；由得有(x,y)BC(x,y)AC(x,y)BC或。根據(jù)并集的定義，由(x,y)(AC)(BC)(AB)C(AC)(BC)(AC)(BC)(AB)C。(x,y)(AC)(BC)(x,y)AC(x,y)BC或(x,y)ACAyC且(x,y)BCxByCxAxBxAB或得，得x得且(x,y)(AB)C(AC)(BC)(AB)C。(AB)C(AC)(BC)。(AB)C(AC)(BC)．(AB)C(AC)(BC)xAByC(x,y)(AB)C且xAB得xA且xBxA和yC得(x,y)ACxB和yC得(x,y)BC。(x,y)AC(x,y)BC(x,y)(AC)(BC)或有(AB)C(AC)(BC)。(AC)(BC)(AB)C(x,y)(AC)(BC)(x,y)AC(x,y)BC(x,y)AC且AyC且(x,y)BCyCxBxAxBxAB且得，得x且得(x,y)(AB)C(AC)(BC)(AB)C。(AB)C(AC)(BC)。(AB)C(AC)(BC)．(AB)C(AC)(BC)60(x,y)(AB)CxAByC且xABAxB且且或xBxA。得xAxBxAyC(x,y)ACxByC和(a)當(dāng)x且和得或得有(x,y)BC(x,y)AC(x,y)BC。根據(jù)對(duì)稱差的定義，由(x,y)(AC)(BC)(AB)C(AC)(BC)；和B且xAxAyC(x,y)ACxByC和（xb和得得得(x,y)BC(x,y)AC(x,y)BC，由對(duì)稱差的定義知，由(x,y)(AC)(BC)(AB)C(AC)(BC)(AC)(BC)(AB)C。(x,y)(AC)(BC)(x,y)AC(x,y)BC且或?qū)τ?，則(x,y)AC(x,y)BC且。(x,y)AC(x,y)BC且(x,y)ACxAyC得且（a(x,y)BCyCxBxAxB和得且得。根據(jù)對(duì)稱差的定義，由xABAB，再由x(AC)(BC)(AB)CAByC(x,y)(AB)C得，因此且。(x,y)AC(x,y)BC且(x,y)BCxByC得且（b。根據(jù)對(duì)稱差的定義，由(x,y)ACyCxAxAxB和得且得xBAAB，再由x(AC)(BC)(AB)CAByC(x,y)(AB)C得，因此且。(AB)C(AC)(BC)。設(shè)ZZZZ((a,b),(c,d（2R是ZR11acbdR161Z((a,b),(c,dR（3）設(shè)R是Z上的二元關(guān)系，使得屬于，當(dāng)且僅當(dāng)22(ac)bd)10RRRRRRRRR，和22，，212121212解Z）Z）1R1）R2RR112RR112RR112RR1121A(2,2),(4,4)}）R1解1234）R2解1234{(）R362解1234AR(2,2),1R2R(2,2),3，和RRR123解R1R2R3Z{(i,i)|i,iZ且|ii）R1121212解R1{(i,i)|i,iZii）R2121212解R2{(i,i)|i,iZ且|i|i|}）R3121212解R3A(ab)(c)(d)A解（a（b63（c（d331212331212AARR解R(2,2),(4,4)}~~RR12RRRR，2，R，。12112解R2(2,2),(2,3)}1RR12~R1~R(3,2)}2~~RR{(12A設(shè)A，R{(i,j)|jiji/1R{(i,j)|ij264RRR(RR)R））R）1221121解R{(0,0),1R{(2,0),2RR12RR{(2,0),21(RR)R(2,2)}121設(shè)，R是AR和R是A若R1212RRRRRRRR。12211221x,y,(x,y)RRzA(x,z)R且212(z,y)RRR(y,z)R(z,x)R為和是A且11212(y,x)RRRRRR(y,x)RR121222121yA(y,y)R(y,x)RR且和R是A111112(x,y)R(y,y)R(x,y)RRRRRR。且1112122112RRRR。1221~{(x,y)|(y,x)}R。RRAR~~R,RRR解~若RR~~xA(x,x)R(,)有xxR以R是R~若R,則R~x,yA(x,y)R(y,x)RR(x,y)R，65~~R(y,x)R~若RR~~x,yA(x,y)R(y,z)R且(y,x)R(z,y)R且R~~R(z,x)R(x,z)RRRR，R，RAR，R都有傳遞性，試問(wèn)R，12121212RR12解R）R12RRR和R設(shè)A，，1212RR12R）R12x,y,zA(x,y)RR(y,z)RR(x,y)R(y,z)R，且121211(x,y)R(y,z)R(x,z)R(x,z)RR和R且且221212(x,z)RRRR1212R）R12R2)}RRR設(shè)A，，和。1212RR12a,b,}Rt(R),設(shè)S{(a,b),(a,c),(c,b）R解{(a,b),(a,c),(c,bRRRR{(a,b)}266R3R(R)RR{(a,b),(a,c),(c,b)}。2Rt2{(a,b),(b,c),(c,c）R解{(a,b),(b,c),(c,cRR2RR{(a,c),(c,c)}R3R2R{(a,c),(c,c)}R2t(R){(a,b),(a,c),(b,c),(c,c)}。{(a,b),(,a),(c,c）R解{(a,b),(,a),(c,cRR2RR{(a,a),(b,b),(c,c)}{(,),(,),(,)}R3R2RabbaccRt(R){(a,a),(b,b),(c,c),(a,b),(b,a)}。{(a,b),(,c),(c,a）R解{(a,b),(,c),(c,aRR2RR{(a,c),(b,a),(c,b)}R3R2R{(a,a),(b,b),(c,c)}{(,),(,),(,)}R4R3RaabccaRt(R){(a,a),(b,b),(c,c),(a,b),(b,c),(c,a),(a,c),(b,a),(c,b)}。設(shè)和RRRAR1212(R)r(R)）r12(R)RRr(R)RR，r為22221267r(R)r(R)r(R)R為則R12211r(R)r(R)。12(R)s(R)）s12~x,yA(x,y)s(R)RR(x,y)R或1111~~(x,y)R(x,y)RRR(x,y)RRRs(R)11122222~s(R)s(R)(x,y)R(y,x)RRR(y,x)R1211122~~(x,y)RRRs(R)s(R)s(R)。222212(R)s(R)s。12(R)t(R)）t12(R)RRt(R)RRt為，222212t(R)t(R)t(R)R為則R12211t(R)t(R)。12設(shè)R和RA1）r2(RR)r(R)r(R)1212r(RR)(RR)(R)(R)r(R)r(R)1）s212A1A2A12(RR)s(R)s(R)1212~~~~s(RR)(RR)(RR)(RR)(RR)s(R)s(R)121212112212(RR)t(R)t(R)（3）t，并舉一個(gè)反例說(shuō)明一般情況下1212t(RR)t(R)r(R)。1212對(duì)于x,yA(x,y)t(R)t(R)(x,y)t(R)，若，則根據(jù)并的定義知或12168(x,y)t(R)。2(x,y)t(R)(x,y)RRR(x,y)RRR若213111112(x,y)(RR)(RR)(RR)t(RR)而有23，因此12121212t(RR)t(R)r(R)；1212(x,y)t(R)(x,y)RRR(x,y)RRR若223222212(x,y)(RR)(RR)(RR)t(RR)而有23，因此12121212t(RR)t(R)r(R)。1212(RR)t(R)r(R)t。12122)}Rt(R)2)}t(R)，令R，1212(RR)(2,2)但tt。12(RR)t(R)r(R)。1212R(2,2),(4,4),A/R?，A解R[2]RR[4]RA/R{,,[4]}。RRRRRA，是AR是ARSRSR，S則。RS是A?RA/R?69）{(R?A1解(2,2),(4,4),(3,4),(4,3)}/R）R）A）R(2,2),(4,4),(2,4),(4,2),1{A,A,,A}AAA12nx,yA(x,y)Riin,xyA明R是A，ixA為AA{,,,}Aiin,xxA，A為12ni(x,x)R則。x,yA(x,y)Riin,xyAiy,xA(y,x)Rix,y,zA(x,y)R(y,z)Ri,jinjn且，,yAy,zAAAx,zA(,)xzR。有x且ijijiR是AA，是RAA/RR?解R(2,2),(4,4),{A,A,,A}1..AB設(shè)Ain12ni{AB,AB,,A}AB12ninABi{A,A,,A}1..AAn為Ai12ni70ABAB。i{A,A,,A}AAAA（3為A12n12nn(AB)(AB)(AB)(AB)(AAA)BABi12n12ni1i,jn()()AB()()xABABABijijABxABxAxAAA此。x且且ijijij{A,A,,A}AAABAB。為A12nijij{AB,AB,,A}ABabi,cdiC*，12nCCC*，C是復(fù)數(shù)集。在*上定義二元關(guān)系：(abi,cdi)Ra*c0是C*R|C/R?*aC*a*a0(abi,abi)R。abi,cdiC(abi,cdi)R*0ac有*c*a0(cdi,abi)R。abi,cdi,efiC*(abi,cdi)R(cdi,efi)R，，*c0c*e0a*c*c*e0，a*e0(abi,efi)R。則a是C*R|C/R2*RAAA，ab(a,b)R是ARaAbA，使得(a,b)RA上R71(a,b)R,a)R得RA(a,b)R,a)R(a,a)RR和得RAR是A設(shè)RAT是A(a,b)T(a,b)和b,a)T是AxA(a,a)RRA(a,a)R(a,a)R(a,a)T和由即且。x,yA(x,y)T(x,y)T(x,y)R(y,x)R得且，(y,x)R(x,y)R(y,x)T且。x,y,zA(x,y)T(y,z)T(x,y)R(y,x)R，且(x,y)R且(z,y)R(y,z)R，RA(y,z)R(x,z)R得(z,y)R(y,x)R(z,x)R且得(x,z)R(z,x)R(x,z)T。和得綜上所述，T是A{(a,b)|c,使得(a,c)R,(c,b)}設(shè)R是ASR是AASxA(x,x)R(x,x)R為R是A(x,x)R(x,x)S得。和x,yA(x,y)SzA(x,z)R，使得對(duì)于，若，則由已知條件知，且；72(z,y)RRAR(z,y)R(y,z)R是得(x,z)R(z,x)R得(y,z)R(z,x)R(y,x)S且得。由且x,y,zA(x,y)S(y,z)Sx,yA11(x,x)R(x,y)R(y,y)R(y,z)R(x,x)R，且，R11111(x,y)R(x,y)R(y,y)R(y,z)R(y,z)R且得且得111(x,y)R(y,z)R(x,z)S和得。AS(a,b)(a,c)和屬設(shè)RAR于R(b,c)R。(a,b)R(a,c)R，RRA(a,b)R,a)R得RA,a)R(a,c)R和得b,c)R。x,yA(x,y)R,R是A(x,x)R(x,y)R(x,x)R(y,x)RR是A和得x,y,zR(x,y)R(y,z)R，R是A(x,y)R(y,x)R得(y,x)R(y,z)R(x,z)R和得。R是A{A,A,,A}是A的子集的集合，使得ij設(shè)RA時(shí)12nA，并且和在子集中的一個(gè)，當(dāng)且僅當(dāng)有序?qū)b在R中。證明ab(,)有Aij{A,A,,A}是A12n73i,jnijAA，AAjiiiA則Aij{A,A,,A}1..AA是Ain，。12nin{A,A,,A}AAxA是A12nii1為R(x,x)R(x,x)Rin,得xxA，innnAAAAAA。xiiiii1i1i1i,jnAAcA且cAAA則aAijijijiAbAbAaAcA(,)cA得acR和且a，且和jjiiijbA得(c,b)R(a,c)R(c,b)R(a,b)R和得R。jabAbAbAAA有和aiiiij{A,A,,A}是A12n設(shè)AA(2,2),(4,4)}R1RR23(2,2),(2,3),(2,2),(4,4),R4解R1123474R2R3R4解(a)b)(c)，設(shè)D{xN|x|x,yD(x,y)R，x|y。?R(D(30),R)解）DR(2,2),(6,6),(2,6),(2,30),(D(30),R)7556321(,)aA1Aa(a,a)得，設(shè)221(a,a)aaA則aAa122121(a,a)aa(a,a)素a3A322332aaaaaa(a,a)32313112(a,a)(a,a)aaaa和得,A21122121(,)|An設(shè)當(dāng)n1knk1na,a,,a,a}Aa}Aa,a,,a}|，An。A12kk1k112k(,)(,)d。(,)Addaglb(,)與ak1k1glb(d,a)(,)Ak176XR是XRR為XS為XS為XXxXS為XS(x,x)S(x,x)S有。Xx,yX(x,y)S且(y,x)S(x,