# 集合练习题2

ê?c

??ù?

8??K?ù
o d 2011.09
1. ‰?

ên ≥ 3,

A1 , A2 , · · · , A2n ?8?{1, 2, · · · , n}
2n

üü??

??f8, PA2n+1 = A1 . ?

i=1

|Ai ∩ Ai+1 | |Ai | · |Ai+1 |

???. )‰ é?? i, e|Ai ∩ Ai+1 | = 0, K
|Ai ∩ Ai+1 | = 0. |Ai | · |Ai+1 |

±eb |Ai ∩ Ai+1 | = 1. dAi ∩ Ai+1 ? Ai 9Ai ∩ Ai+1 ? Ai+1 ,
2n

max(|Ai |, |Ai+1 |) ≥ 2, u?
2n

i=1

|Ai ∩ Ai+1 | ≤ |Ai | · |Ai+1 |

i=1

1 ≤ max(|Ai |, |Ai+1 |)

2n

i=1

1 = n. 2

?? ???± , ~XA1 = {1}, A2 = {1, 2}, · · · · · · , A2i?1 = {i}, A2i = {i, i + 1}, · · · · · · , A2n?1 = {n}, A2n = {n, 1}.
2. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A1 , A2 , · · · , Ak ?S (1) |Ai | = 5 (1 ≤ i ≤ k ) ; (2) |Ai ∩ Aj | ≤ 2 (1 ≤ i < j ≤ k ) .

f8, ÷v

?k ???. )‰? ?10 × k

L?, ??1i1, 1j
aij = 1, ei ∈ Aj , 0, ei ∈ Aj .
k

?

??

i = 1, 2, · · · , 10; j = 1, 2, · · · , k.
10

u?L?1i1
|Aj |L?8?Aj ?

???li =
j =1

aij L?iáuA1 , A2 , · · · , Ak ?li ?8?,
10

1j

???
i=1

aij =

?

?ê, d??^?(1) k
i=1 10 10 k k

aij = |Aj | = 5, ¤±
10 k

li =
k=1 i=1 j =1

aij =
j =1 i=1

aij =
j =1

|Aj | = 5k.

(1)

er ∈ Ai ∩Aj , Kò{Ai , Aj , r}|¤3 |, ù?n | ?êP?S . ???, ?ráuAi , A2 , · · · , Ak ?
10

l r ? 8 ? , ? / ¤ C2 lr ??r

n

|, ¤ ±S =
r =1

C2 lr . ,???, é??Ai , Aj (1 ≤ i < j ≤ k )

1

k|Ai ∩ Aj |? ?áuAi ∩ Aj , ?/¤|Ai ∩ Aj |??Ai ?Aj
10

n |, ¤±S =
1≤i<j ≤k 10 2 lr ? r =1 r =1 10

|Ai ∩ Aj |, u?

| Ai ∩ Aj | =
1≤i<j ≤k r =1

C2 lr =

1 2

lr

.

d?(1) 9…?? ?,
k (k ? 1) = 2C2 k ≥
1≤i<j ≤k

? 1?1 |Ai ∩ Aj | ≥ 2 10

10

2

10

? li ? = 5 1 [(5k )2 ? 10 · 5k ] = k (k ? 2), 20 4

li
i=1

?
i=1

¤±, k ≤ 6. ?g, e 6?8?÷vK

^?:

A1 = {1, 2, 3, 4, 5}, A2 = {1, 2, 6, 7, 8}, A3 = {1, 3, 6, 9, 10}, A4 = {2, 4, 7, 9, 10}, A5 = {3, 5, 7, 8, 10}, A6 = {4, 5, 6, 8, 9}.

n???, ¤?k ????6. )‰ ?)‰?? E6?f8A1 , A2 , · · · , A6 ÷v^?,
k

¤? ????

u6.
35 ? 1 +1 = 10

ek ≥ 7, K
i=1

|Ai | = 5k ≥ 35,

S ?–

k??

?a–

áuA1 , A2 , · · · , Ak ?

4???8?. ?” A3 ∩ A4 | ≥ 1, dN?
4

a ∈ Ai (i = 1, 2, 3, 4) , u?é??1 ≤ i < j < t ≤ 4, |Ai ∩ Aj ∩ At | ≥ |A1 ∩ A2 ∩

n,
4

10

= ≥

|S | ≥
i=1

Ai =
i=1

| Ai | ?
1≤i<j ≤4

|Ai ∩ Aj | +
1≤i<j<t≤4

|Ai ∩ Aj ∩ At | ? |A1 ∩ A2 ∩ A3 ∩ A4 |

3 4 × 5 ? C2 4 × 2 ? (C4 ? 1) × 1 = 11,

g?. ¤±k ≤ 6. n???¤?k
3.

????6.

A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, · · · , n}, 3A?? 3?ê, 3B ? ü?ê, |¤?k5? ? 8?Ai (i = 1, 2, 3, · · · , 20), ? |Ai ∩ Aj | ≤ 2, 1 ≤ i < j ≤ 20. ?n ? ?. )‰ ·?ky?, B ?z? ?3?f8Ai (i = 1, 2, · · · , 20) ?–??y4g. e?,, b?B ? , ?x3??f8Ai (i = 1, 2, · · · , 20) ??y k ≥ 5g, ? ?x k?f8, §? ?A?3k ≥ 15? 15 ? 1 ?, d?T K, A?– k1? ?, ?y , 3ùk?f8?– ?y + 3g. ???x, 6 y 3?f8?Ar , As , At (1 ≤ r < s < t ≤ 20) , KA\{y }?5? ?3Ar , As , At ??y 2 × 3 = 6g. u?7k?? ?z ?y üg, ????x, y , z ü?f8, ù???^?|Ai ∩ Aj | ≤ 2 (1 ≤ i < j ≤ 20) g?. d??y??B ?z? ?3A1 , A2 , · · · , A20 ?–??y4g, B ?? ?3A1 , A2 , · · · , A20 ?? 40 = 10, ¤±, n ≥ 10 + 6 = 16. y odê?2 × 20 = 40, ¤±|B | ≥ 4 n = 16?, ???Xe20?8?: {1, 2, 3, 7, 8}, {1, 2, 4, 12, 14}, {1, 2, 5, 15, 16}, {1, 2, 6, 9, 10}, {1, 3, 4, 10, 11}, {1, 3, 5, 13, 14}, {1, 3, 6, 12, 15}, {1, 4, 5, 7, 9}, {1, 4, 6, 13, 16}, {1, 5, 6, 8, 11}, {2, 3, 4, 13, 15}, {2, 3, 5, 9, 11}, {2, 3, 6, 14, 16}, {2, 4, 5, 8, 10}, {2, 4, 6, 7, 11}, {2, 5, 6, 12, 13}, {3, 4, 5, 12, 16}, {3, 4, 6, 8, 9}, {3, 5, 6, 7, 10}, {4, 5, 6, 14, 15}. 2

ê, 8?S = {1, 2, · · · , n}, é?? k??ê8?A?B , ?|A?S | + |B ?S | + |C ?S | ? ?, ??C = {a + b|a ∈ A, b ∈ B }, X ?Y = {x|xT?áuX ?Y ? ??}, |X |L?k? 8?X ??ê. )‰ ¤? ? ??n + 1. ?k, A = B = S , ??|A?S | + |B ?S | + |C ?S | = n + 1. e?y?: l = |A?S | + |B ?S | + |C ?S | ≥ n + 1. PX \Y = {x|x ∈ X, x ∈ Y }. w,, l = |A\S | + |B \S | + |C \S | + |S \A| + |S \B | + |S \C |. u?, ? ?y?: (I) |A\S | + |B \S | + |S \C | ≥ 1; (II) |C \S | + |S \A| + |S \B | ≥ n. ky(I) . ???, e|A\S | = |B \S | = 0, KA, B ? S . 1??U?C ? ?, =S \C | ≥ 1. 2y(II) . eA ∩ S = ?, K|S \A| ≥ n, (?¤á. eA ∩ S = ?, KA ∩ S ???? ???n ? k (0 ≤ k ≤ n ? 1) . K |S \A| ≥ k. (1)
4.

n? ‰ ?

,???, éi = k + 1, k + 2, · · · , n, ??i ∈ B (d?i ∈ S \B ) , ??i ∈ B (d?n ? k + i ∈ C , =n ? k + i ∈ C \S ) . ¤±, |C \S | + |S \B | ≥ n ? k. (2) d?(1) (2) = (II) . n???, l ≥ n + 1. ¤±, ¤??
5. ?¤k

??n + 1. f8A?B

êk, ? 8?X = {1994, 1997, 2000, · · · , 1994+3k}?±?¤ü??? ?8, …A? ? ??B ? ? ? 9 . )‰ X Skk + 1 ? ê, ùk + 1 ? ê??P?S , ??K?,
1 S = 1994 + 1997 + 2000 + · · · + (1994 + 3k ) = 1990(k + 1) + (k + 1)(3k + 8). 2

(1)

PB S ?

ê???x, KAS ?

1 ê???9x, u?10x = 1990(k + 1) + (k + 1)(3k + 8), = 2 1 (k + 1)(3k + 8). 20 (2)

x = 199(k + 1) +

l(2) ?±? , (k + 1)(3k + 8)7?20 ê. l ?k???Uk = 5m ? 1 (m ∈ N+ ) , …m(15m + 5)?4 ê, @o, m = 4t?m = 4t ? 1 (t ∈ N+ ) . ??1: m = 4t, k = 20t ? 1 (t ∈ N+ ) . X S?k20t? ê, òX ? ê U l ? ^Sü , l1994?z20??U ê? ???? üêy8B , ?m18êy8A, ù A?B , w,÷v K8??. ??2: m = 4t + 1, k = 20t + 4 (t ∈ N+ ) . 2|^(2) , k
x = 5(4t + 1)(3t + 200). (3)

qX ?z?

ê?±3{2, ^|B |L?8?B ?

? ?ê, @o, 2(?(3) , k
(mod 3). (4)

2|B | ≡ x ≡ t + 1

u?,
|B | ≡ 2t + 2 (mod 3). (5)

3

ê?? ?(1990 + 436) + (1990 + 433) + · · · + (1990 + 400) = 1990 × 13 + 418 × 13 < 1990 × 13 + 6175. X S? 16? ê???(1990 + 4) + (1990 + 7) + · · · + (1990 + 49) = 1990 × 16 + 424 > 1990 × 16 + 205. d±???, t = 7?, ??3÷vK8?? f8A?B . t = 8?, x = 1990 × 8+1140. -B = {1994, 1997, 2000, · · · , 2036, 2039, 2210, 2486}, ?? k18? ê. B S ? ê??T?1990 × 18 + 1140. -A = X \B , u? t = 8?, 8?X k÷vK? ?). XJ,? êk ÷ v ? ?, @ o é u êk + 20, ??òX ?#V\ 20?êUc?¤? ? {, ???? üê8\B , ?{18?ê8\A, @o, k + 20? ê?÷vK?. u?, k = 20t + 4?… êt ≥ 8?, 8?X k÷vK? ?). n?¤?, ¤? ê?k = 20t ? 1 (t ∈ N+ ) 9k = 20t + 4 (t ≥ 8) .
6. m?n?‰?

t = 7?, d(3) kx = 1990 × 13 + 6175 = 1990 × 16 + 205. ùp, X S??

13?

? u1

ê, a1 < a2 < · · · < am ??
|T | ≤ 1 + am ? a1 , 2n + 1

ê. y ?: ? 3

ê8

??f8T , ?

??ê

…éz?i ∈ {1, 2, · · · , m}, ?kt ∈ T 9s ∈ [?n, n], ? ai = t + s. )‰ -a1 = a, am = b, ?‘{?{b ? a = (2n + 1)q + r, ??q, r ∈ Z…0 ≤ r ≤ 2n. b?a {a + n + (2n + 1)k |k = 0, 1, · · · , q }, K|T | = q + 1 ≤ 1 + , …8? 2n + 1
B = {t + s|t ∈ T, s = ?n, ?n + 1, · · · , n} = {a, a + 1, · · · , a + (2n + 1)q + 2n}.

T =

5? a + (2n + 1)q + 2n ≥ a + (2n + 1)q + r = b, ?dz?ai ?3B ?, l (?¤á. ¤k ê|¤ 8?, S ?T ??f8, ??vk??ê?,??ê ê. @oS ???k? ? ?? )‰ 5? 2004 = 22 × 3 × 167, KT = {2a 3b 167c |0 ≤ a ≤ 200, 0 ≤ b, c ≤ 100}. S = {2200?b?c 3b 167c |0 ≤ b, c ≤ 100}. éu0 ≤ b, c ≤ 100, k0 ≤ 200 ? b ? c ≤ 200, ¤±, S ? k1012 ? ?. e?y?: S ?vk??ê?,??ê ê. b 2200?b?c 3b 167c ?2200?i?j 3i 167j ê, K ? ? ? ? 200 ? b ? c ≥ 200 ? i ? j, b ≥ i, ? ? ? c ≥ j,
7. T ?d2004100

=b = i, c = j . S ?vk??ê?,??ê ê. 2 2^?y{y?: ÷v^? S ???k101 ? ?. U ?T ???L1012 ? ? f8. ? ??k1012 ?p? (b, c), ¤±, d?T K?7kü? ?u1 = 2a1 3b1 167c1 , u2 = 2a2 3b2 167c2 ? b1 = b2 , c1 = c2 , a1 = a2 . u?, a1 > a2 ?, u1 ?u2 ê; a1 < a2 ?, u2 ?u1 ê. ?d, f 8U ?÷v^?. ¤±, S ???k1012 ? ?.
8. X = {1, 2, · · · , 2011}. ?? u + v ?2

êm, ·???: éX

????m f8W , ??3u, v ∈ W

(u?v ?±??) , ?

??. )‰ òX ?¤±e5?f8?1? : 2001 = 1024 + 977 ≥ x ≥ 1024 ? 977 = 47, 46 = 32 + 14 ≥ x ≥ 32 ? 14 = 18, 17 = 16 + 1 ≥ x ≥ 16 ? 1 ≥ 15, 14 = 8 + 6 ≥ x ≥ 8 ? 6 = 2, x = 1. ? E ? ? ? K ? ? ? ? ÷ v … q ? ? ? ? ~ f, ù ? f 8 ? U ?2 ? ? ? ? … z é r ê{2 + a, 2r ? a}??Uk???38?. -Y = {2001, 2000, · · · , 1025} ∪ {46, 45, · · · , 33} ∪ {17} ∪
4

{14, 13, · · · , 9} ∪ {1}, Kk|Y | = 998, …é??u, v ∈ Y , u + v ???2
r r r +1

??. ???, u, v ∈ Y ?, ? ” u ≥ v …k2 < u ≤ 2 + a < 2 , ?? r?O ?10, 5, 4, 3?, ?A a??g?977, 14, 1, 6. (1) e2r < v ≤ u, K2r+1 < u + v < 2r+2 , u + v ?U?2 ??; (2) e1 ≤ v < 2r , K 2r < u ≤ 2r + a, 1 ≤ a < 2r ?, 1 ≤ v ≤ 2r ? a. u?2r < u + v < 2r+1 , ùL ?u+v ??U?2 ??. ¤±, f8Y ???üê?????2 ??. ?¤? ? êm ≥ 999. òX y?¤e 999?p?? f8: Ai = {1024 ? i, 1024 + i}, i = 1, 2, · · · , 977; Bj = {32 ? j, 32 + j }, j = 1, 2, · · · , 14; C = {15, 17}; Dk = {8 ? k, 8 + k }, k = 1, 2, · · · , 6; E = {1, 8, 16, 32, 1024}. éuS ????999 f8W , eW ∩ E = ?, Kl??? ?? ? 2 ??2 ??; eW ∩ E = ?, KW ? 999? ??áuc?998?2 f8. d?T K?W ?7k?? u?v , áu?? f8. w,, u + v ?2 ??. n???, ¤? ? êm = 999.
9. b) | ab. S = {1, 2, · · · , 50}. ??

êk , ?S

??k

f8???3ü??? êa?b, ÷v(a +

)‰ éu÷v^?(a + b) | ab a, b ∈ S , Pc = (a, b), u?a = ca1 , b = cb1 , ??a1 , b1 ∈ N …(a1 , b1 ) = 1. ? k c(a1 + b1 ) = (a + b) | ab = c2 a1 b1 ,
+

=(a1 + b1 ) | ca1 b1 . ??(a1 + b1 , a1 ) = (a1 + b1 , b1 ) = (a1 , b1 ) = 1, ¤±
(a1 + b1 ) | c. (1)

q??a, b ∈ S , ¤±a + b ≤ 99, =c(a1 + b1 ) ≤ 99. d(1)?3 ≤ a1 + b1 ≤ 9. dd??, S ÷v^ ?(a + b) | ab ¤kêéXe:
a1 + b1 = 3 a1 + b1 = 4 a1 + b1 = 5 a1 + b1 = 6 a1 + b1 = 7 a1 + b1 = 8 a1 + b1 = 9 : : : : : : : (6, 3), (12, 6), (18, 9), (24, 12), (30, 15), (36, 18), (42, 21), (48, 24); (12, 4), (24, 8), (36, 12), (48, 16); (20, 5), (40, 10), (15, 10), (30, 20), (45, 30); (30, 6); (42, 7), (35, 14), (28, 21); (40, 24); (45, 36).

k23é. -M = {6, 12, 15, 18, 20, 21, 24, 35, 40, 42, 45, 48}, K|M | = 12…??23?êé?z?êé??– ?kM ? ?? ?. ?d, e-T = S ? M , K|T | = 38…T ? ??üê??÷vK? ??. ¤±, ¤? ? êk ≥ 39. 5? e 12?êé
(6, 3), (12, 4), (20, 5), (42, 7), (24, 8), (18, 9), (40, 10), (35, 14), (30, 15), (48, 16), (28, 21), (45, 36)

p?? …?÷vK? ??. ¤±, éuS ??39 f8, §?'S 11? ?, áu??12?êé? 11é, l 7k12é? ?éáuù?39 f8. n???, ¤? ? êk = 39.
10. ?÷v x?y y?z z?u u?x + + + > 0, …1 ≤ x, y, z, w ≤ 10 x+y y+z z+u u+x

ù11? ?–?

¤ko kS ê|(x, y, z, u)

?ê.
5

a?b b?c c?d d?a )‰ f (a, b, c, d) = + + + . PA = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f (x, y, z, u) > a+b b+c c+d d+a 0}, B = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f (x, y, z, u) < 0}, C = {(x, y, z, u)|1 ≤ x, y, z, u ≤ 10, f (x, y, z, u) = 0}. w,Card(A) + Card(B ) + Card(C ) = 104 . · ? y ?Card(A) = Card(B ). é z ? ?(x, y, z, u) ∈ A, ? ?(x, u, z, y ). (x, y, z, u) ∈ A ? y?z z?u u?x x?u u?z z?y y?x x?y + + + > 0 ? + + + < 0 ? f (x, y, z, u) > 0 ? x+y y+z z+u u+x x+u u+z z+y y+x f (x, u, z, y ) < 0 ? (x, u, z, y ) ∈ B . xz ? yu xz ? yu XO?Card(C ). (x, y, z, u) ∈ C ? = ? (z ?x)(u?y )(xz ?yu) = 0. (x + y )(z + u) (y + z )(u + x) C1 = {(x, y, z, u)|x = z, 1 ≤ x, y, z, u ≤ 10}, C2 = {(x, y, z, u)|x = z, y = u, 1 ≤ x, y, z, u ≤ 10}, C3 = {(x, y, z, u)|x = z, y = u, xz = yu, 1 ≤ x, y, z, u ≤ 10}. ??÷vab = cd, 1 ≤ a, b, c, d ≤ 10 üü?? ?So |?k1 × 6 = 2 × 3, 1 × 8 = 2 × 4, 1 × 10 = 2 × 5, 2 × 6 = 3 × 4, 2 × 9 = 3 × 6, 2 × 10 = 4 × 5, 3 × 8 = 4 × 6, 3 × 10 = 5 × 6, 4 × 10 = 5 × 8. ÷vx = z , y = u, xz = yu o | 90?, ÷vx = u, y = z , x = z o | 90?, ÷vx = u, y = z , x = z o | 90?, u?Card(C3 ) = 4 × 2 × 9 + 90 + 90 = 252. qCard(C1 ) = 1000, Card(C2 ) = 900, KCard(C ) = 2152. ?dCard(A) = 3924.

6