专题1.1集合七大题型（原卷版）_11.4-2_2026-11-04

      
      
专题1.1 集合【六大题型】

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 1   !"#
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§ 1¨2024 Y©·£ª·«¬­𝐴= 𝑥|𝑚𝑥2 + 2𝑥+ 2 = 0 *®#!"E“
 m &¯°±
²p¨    ­
As{𝑚|𝑚≠0}
Bs 𝑚|𝑚< 1
2
 
Cs 𝑚|𝑚< 1
2 且𝑚≠0
 
Ds 𝑚|0 < 𝑚< 1
2
 
S] 1-1¨2023·³´µ¶··¸¹º­»r𝑃= {𝑛|𝑛= 2𝑘−1,𝑘∈𝑁∗,𝑘≤10}E𝑄= {2,3,5}E“
      
      
𝑇= {𝑥𝑦|𝑥∈𝑃,𝑦∈𝑄} !"&#
p¨    ­
As30
Bs28
Cs26
Ds24
S] 1-2¨2023·¼½´¾··¸¹º­34𝐴∗𝐵= {𝑧∣𝑧= 𝑥𝑦,𝑥∈𝐴,𝑦∈𝐵}E¿𝐴= {−1,0,1}E
𝐵= {−1,1,3}E“𝐴∗𝐵 !"&#
p¨   ­
As4
Bs5
Cs6
Ds7
S] 1-3¨2023·³À··¸¹º­ U * 71 #!"E𝑆,𝑇⊆𝑈•Á* 14E28 #!"E“∁𝑆∪𝑇(𝑆∩𝑇)
&!"#
ÂÃF¨    ­
As14
Bs30
Cs32
Ds42
 2  %&#
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§ 2¨2024·ÄÅ·Y·­»r𝐴= {−1,0,1}E𝐵= {𝑥|𝑥= 𝑚𝑛 ,𝑚∈𝐴,𝑛∈𝐴}E“ B &8%#
F¨    ­
As4
Bs7
Cs8
Ds15
S] 2-1¨2024·£ª··¸¹º­»r{1,0} ⊆𝐵￿{−1,0,1,2}E“ÆÇÈÉ&𝐵&#
p¨    ­
As3
Bs4
Cs5
Ds6
S] 2-2¨2024·£ª··¸¹º­»r𝐴= {−1,0,1}E𝐵= {𝑧|𝑧= 𝑥+ 𝑦+ 1,𝑥∈𝐴,𝑦∈𝐴 }E“𝐵&
8%#
p¨    ­
As8
Bs16
Cs31
Ds63
S] 2-3¨2024·£ª··¸¹º­»r𝐴= (𝑥,𝑦)|𝑥,𝑦∈Z,且𝑥𝑦= 4 E𝐵= {(𝑥,𝑦)|𝑥≤𝑦}E“𝐴∩𝐵&
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p¨    ­
As3
Bs4
Cs8
Ds16
 3  '(&)*&+,$
§ 3¨2024·£ª··¸¹º­»r𝐴= {𝑚+ 2,1,4},𝐵= {𝑚2,1}E𝐵⊆𝐴E“
𝑚= ¨    ­
As−1
Bs−2~ 0
Cs−2
Ds2
S] 3-1¨2023·£ª·8­¿𝐴= {0,−𝑎}E𝐵= {1,𝑎−2,2𝑎−2}E𝐴⊆𝐵E“𝑎= ¨    ­s
As2
Bs1
Cs
2
3
Ds−1
S] 3-2¨2024·¼½ÊË·t·­»r𝐴= {𝑥|1 < 𝑥< 2024}E𝐵= {𝑥|𝑥< 𝑎}E𝐴⊆𝐵E“
 a &
¯°±²F¨   ­
As(2024, + ∞)
Bs[2024, + ∞)
Cs(−∞,2024]
Ds(−∞,2024)
      
      
S] 3-3¨2024·ÌÄÍηt·­¿𝐴= {1,−𝑎},𝐵= {0,3−𝑎,3𝑎−8}E𝐴⊆𝐵E“𝑎= ¨    ­
As0
Bs1
Cs2
Ds3
 4  &-./.012
§ 4¨2024·Ïз·¸¹º­»r𝐴= {𝑥∈N|𝑥> 1}E𝐵= {𝑥|0 < 𝑥< 4}E“𝐴∩𝐵= ¨    ­
As{𝑥|1 < 𝑥< 4}
Bs{𝑥|𝑥> 0}
Cs{2,3}
Ds{1,2,3}
S] 4-1¨2024·Ñ´Ò³·[·­¿𝐴= {0,1,2},𝐵= {3,𝑚}E𝐴∩𝐵= {2}E“𝐴∪𝐵= ¨    ­
As{0,1,2,3}
Bs{0,1,2}
Cs{1,2,3}
Ds{2,3}
S] 4-2¨2024·¼½ÓÔÕ·t·­»r£𝑈= {𝑥|1 < 𝑥< 6,𝑥∈𝐍 },𝐴= {2,3},𝐵= {2,4,5}E“(∁𝑈𝐴) ∪𝐵
=¨    ­
As{4,5}
Bs{2,3,4,5}
Cs{2}
Ds{2,4,5}
S] 4-3¨2024·£ª··¸¹º­»r𝑈= 𝑍E𝐴= {𝑥|𝑥= 2𝑘+ 1,𝑘∈𝑍 }E𝐵= {𝑥|𝑥= 4𝑘+ 2,𝑘∈𝑍 }E
“{𝑥|𝑥= 4𝑘,𝑘∈𝑍 } = ¨    ­
As∁𝑈(𝐴∩𝐵)
Bs∁𝑈(𝐴∪𝐵)
Cs∁𝑈𝐴∩𝐵
Ds∁𝑈𝐴∪𝐵
 5  '&12*&+,$
§ 5¨2024·ÌÄ··¸¹º­»r𝐴= {−1,𝑎,𝑎+ 2},𝐵= 𝑦|𝑦= 𝑥2−2𝑥,𝑥∈𝐴E𝐴∪𝐵= 𝐴E“𝑎=
¨    ­
As3
Bs2
Cs1
Ds1 ~ 3
S] 5-1¨2024·Ö×ØË·Y·­¿𝑆= {𝑥|𝑥< −1  ~𝑥> 5}E𝑇= {𝑥|𝑎< 𝑥< 𝑎+ 8 }E•
𝑆∪𝑇= RE“
𝑎&¯°±²p¨    ­
As(−∞,−3) ∪(−1, + ∞)
Bs(−3,−1)
Cs(−∞,−3] ∪[−1, + ∞)
Ds[−3,−1]
S] 5-2¨2024·ÙÚ··¸¹º­»r𝐴= {𝑥|𝑥≤𝑎}E𝐵= {𝑥|𝑥≥2𝑎−1}E𝐴∩𝐵≠∅E“‹
𝑎&
¯°±²p¨    ­
As(0,1]
Bs(0,1)
Cs(0,2)
Ds(1,2)
S] 5-3¨2024·ÏÐÛ¶·Y·­»r𝐴= {𝑥|𝑥< 𝑎}E𝐵= {𝑥|−1 < 𝑥< 2}E𝐴∪∁𝑅𝐵= 𝑅E“𝑎&¯
°±²p¨    ­
As(−∞,−1]
Bs(−1,2]
      
      
Cs[−1,2)
Ds[2, + ∞)
 6  &34$
§ 6¨2024·ÜÝÞ·[·­»r𝐴= {1,2}E𝐵= {3,4}E34
𝐴∗𝐵= {(𝑥,𝑦)|𝑥∈𝐴,𝑦∈𝐵 }E“
𝐴∗𝐵&™š%&#
F¨    ­#s
As16
Bs15
Cs14
Ds13
S] 6-1¨2023·Ñ´ßà·[·­3412
𝐴+ 𝐵= {𝑧|𝑧= 𝑥+ 𝑦,𝑥∈𝐴,𝑦∈𝐵}E¿𝐴= {1,2}E𝐵=
{1,2,3}E“𝐴+ 𝐵&ž*!"áâp¨    ­
As14
Bs15
Cs16
Ds18
S] 6-2¨2023·£ª·t·­ã‡ž& Venn ‡ E𝐴.𝐵F™šE34𝐴⊗𝐵päåæ<€
&s𝐴= {𝑥|𝑥= 2𝑛+ 1,𝑛∈𝑁,𝑛≤4}E𝐵= {2,3,4,5,6,7}E“𝐴⊗𝐵= ¨    ­
As{2,4,6,1}
Bs{2,4,6,9}
Cs{2,3,4,5,6,7}
Ds{1,2,4,6,9}
S] 6-3¨2023·£ª··¸¹º­H} AEBE34𝐴−𝐵= {𝑥|𝑥∈𝐴 •𝑥∉𝐵}E»r𝑈=
{𝑥|−3 < 𝑥< 7,𝑥∈Z}E𝐸= {−1,0,2,4,6}E𝐹= {0,3,4,5}E“∁U(𝐸−𝐹) = ¨    ­
As{−2,0,1,3,4,5} Bs{0,1,3,4,5}
Cs{−1,2,6}
Ds{−2,0,1,3,4}
Y.o
1s¨2024·£ª··¸¹º­»r𝐴= {𝑥|𝑥= 3𝑘+ 1,𝑘∈Z}E“烀‹w&F¨     ­.
As−2 ∈𝐴
Bs2023 ∉𝐴
Cs3𝑘2 +1 ∉𝐴
Ds−35 ∉𝐴
2s¨2024·è¶èË··¸¹º­𝐴= {𝑥|2𝑚𝑥−3 > 0,𝑚∈R}Eé 2 ∈𝐴•1 ∉𝐴E“
 m &¯°±
²F¨    ­
As 3
4 , 3
2
Bs 3
4 , 3
2
Cs 3
4 , 3
2
Ds 3
4 , 3
2
3s¨2024·êëìí·Y·­𝐴= 𝑥∈R|𝑎𝑥2−2𝑥+ 1 = 0  î*Y#!"E“
𝑎= ¨    ­
As1
Bs0
Cs2
Ds0 ~ 1
4s¨2024·ïÞ·[·­»r𝑀={1,2,3}E𝑁={0,1,2,3,4,7}E𝑀⊆𝐴⊆𝑁E“ÆÇ𝐴&#
p¨    ­
      
      
As4
Bs6
Cs7
Ds8
5s¨2024·àÐðñ·Y·­»r𝐴= {𝑥||𝑥|≤2 }E𝐵= {𝑥|𝑥−𝑎< 0 }E𝐴⊆𝐵E“𝑎&¯°±²p¨    ­
As(−∞,−2)
Bs(−∞,−2]
Cs(2, + ∞)
Ds[2, + ∞)
6s¨2024·£ª··¸¹º­𝑃= {1,2,3,4,5,6},𝑄= {𝑥|𝑥= 3𝑡−1,𝑡∈𝑃 }E“𝑃∩𝑄= ¨    ­
As{1,3,5}
Bs{2,4,6}
Cs{1,4}
Ds{2,5}
7s¨2024·òó·[·­¿£𝑈= {−1,0,1,2,3}E𝐴= {0,1,2}E𝐵= {−1,0,1}E“(∁𝑈𝐴) ∪𝐵= ¨    ­
As{−1,0,1,3}
Bs{−1,0,1,2}
Cs{0,1,2}
Ds{0,1}
8s¨2024·Ö×·[·­»r𝐴= {1,2,3}E𝐵= {𝑥|𝑥> 𝑎 }E𝐴∩(∁𝑅𝐵) = 𝐴E“
𝑎&¯°±²F¨    ­
As𝑎≥1
Bs𝑎≤1
Cs𝑎≥3
Ds𝑎≤3
[.ô
9s¨2024·£ª··¸¹º­™š A õ*ãçx¥
①𝑥,𝑦∈𝐴E“
𝑥
𝑦∈𝐴”②𝑥,𝑦∈𝐴E“𝑥+ 𝑦∈𝐴ç
ƒö÷ E‹w&*¨    ­
As−1 ∉𝐴
Bs
2022
2023 ∈𝐴
Cs𝑥,𝑦∈𝐴E“𝑥𝑦∈𝐴
Ds𝑥,𝑦∈𝐴E“𝑥−𝑦∈𝐴
10s¨23-24 [©·àëø ·ùú«¬­»r𝐴= {𝑥∣𝑥2−3𝑥−18 = 0}E𝐵= 𝑥|𝑥2 + 𝑎𝑥+ 𝑎2−27 = 0 E
“çƒû ‹w&F¨    ­
As𝐴= 𝐵E“𝑎= −3
Bs𝐴⊆𝐵E“𝑎= −3
Cs𝐵= ∅E“{𝑎∣𝑎< −6~𝑎> 6}
Ds𝐵￿𝐴E“{𝑎∣𝑎< −6~𝑎≥6~𝑎= −3}
11s¨2023·àÐüý·Y·­™š𝑀,𝑁,𝑃ÆÇ
𝑀∩𝑁= 𝑁,𝑀∪𝑃= 𝑃E“¨    ­
As𝑃⊆𝑀
Bs𝑀∩𝑃= 𝑀
Cs𝑁∪𝑃= 𝑃
Ds𝑀∩∁𝑝𝑁= ∅
t.þš
12s¨2024·ÌÄÿзY·­𝑥2−80pĀ£āĂ
E“‹Œ
 x &¯°Ÿ &p      
s
13s¨2024·ÌÄ㥹·Y·­»r𝐴= {−1,2,4}E𝐵= {2,𝑚2}s𝐵⊆𝐴E“
𝑚&¯°p      
s
14s¨2024·£ª··¸¹º­»r𝑀= {−2,−1,0,1}E𝑁= {𝑥|𝑎−3 < 𝑥< 1}E𝑀∩𝑁 * 2 #!"E“

 a &¯°±²F      
s
      
      
¼.Ćć
15s¨23-24 Y©·ÄÅĈĉ·ùú«¬­»r𝐴= 𝑥∈𝑅|𝑎𝑥2 + 2𝑥+ 1 = 0 Eé 𝑎∈𝑅.
(1)𝐴 *•Ċ*Y#!"E7
𝑎Ÿ &𝐵.
(2)𝐴 ċô*Y#!"E7
𝑎&¯°±².
16s¨23-24 Y©·ÏÐČà·čĎ­¿𝐴= {𝑥∣𝑥2−5𝑥+ 6 = 0,𝑥∈𝑅},𝐵= {𝑥∣𝑎𝑥−1 = 0,𝑥∈𝑅}
(1)𝑎=
1
2Eďö÷𝐴'𝐵&)”
(2)𝐵￿𝐴E7𝑎&°Ÿ &𝐶s
17s¨23-24 Yç·¼½ ’·Đď­»r𝐴= {𝑥|𝑚 < 𝑥< 2𝑚+ 1}E𝐵= {𝑥|𝑥≤−3~𝑥> 5}s
(1)đ𝑚= 4ĒE7𝐴∩(∁𝑅𝐵)”
(2)𝐴⊆∁𝑅𝐵E7
𝑚&¯°±²s
18s¨23-24 Y©·ÏÐēÞ·čĎ­»r𝐴= {𝑥|(𝑥−2)(𝑥+ 3)≤0 }E𝐵= {𝑥|𝑎−1 < 𝑥< 𝑎+ 1 }E34
®# PEQ &Ĕ12
𝑃−𝑄= 𝑥|𝑥∈𝑃,且𝑥∉𝑄 s
(1)đ𝑎= 1ĒE7𝐴−𝐵'𝐵−𝐴”
(2)“𝑥∈𝐴”F“𝑥∈𝐵”&G6ÈÉE7
 a &¯°±²s
      
      
19s¨23-24 Y©·ÀĕĖÑ·čĎ­H}‹Œ
𝐴= {𝑎1,𝑎2,⋯,𝑎𝑛}¨𝑛∈𝑁∗E𝑛≥3­ãėĘęé ‘
Y#!".𝑎𝑖(𝑖= 1,2,⋯,𝑛)áĚEěĜ&ž*!"Ÿ &’ĝ<p®#-pš&E•Ğ®#
&ž*!"áâJ\Eğ¤𝐴p“âĠ”.
(1)ö÷{1,2,3,4,5}FġF“âĠ”E/ĢģĤ”
(2)7ĥ
𝐴F“âĠ”.“𝐴 !"#
pĦ
”
(3)𝐴F“âĠ”E7𝐴 !"#
&Âħ°.