专题1.2常用逻辑用语六大题型（原卷版）_11.4-2_2026-11-04

      
      
专题1.2 常用逻辑用语【六大题型】

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 1   !"#$ !%&'
Ì 1Ï2024·JK·ÐÑÒ\]𝑎,𝑏∈𝑅Iz“𝑎= 𝑏= 0”L“|𝑎+ 𝑏| = 0”%Ï    Òx
Axˆ#$ !
Bx#$ˆ !
Cx$ !
Dx ˆ¤ˆ#$ !
Ó_ 1-1Ï2024·ÔՓ·ÑÖ×ØÒ1“𝑥+ 𝑦≤6”L“𝑥≤2IÙ𝑦≤4”%Ï    Ò
Axˆ#$ !
Bx#$ˆ !
Cx#$ !
Dx ˆ¤ˆ#$ !
Ó_ 1-2Ï2023·¾ÚÛÜ·ÐÑÒ¿𝑎,𝑏tÝ
Iz“𝑎> 𝑏> 0”%‹•Þ#$ !LÏ    Ò
Ax 𝑎−1 >
𝑏−1
Bx𝑎2 > 𝑏2
Cx
1
𝑏>
1
𝑎
Dx𝑎−𝑏> 𝑏−𝑎
Ó_ 1-3Ï2023·ßàáâ·ÑÖ×ØÒ¿ AIBICID Lԕ1Iy A L B %#$ˆ !IA L
C %ˆ#$ !ID L B %#$ !Iz D L C %Ï    Ò
Ax#$ˆ !
Bxˆ#$ !
Cx#$ !
Dx ˆ¤ˆ#$ !
 2  () !*#$ !+,

Ì 2Ï23-24 ã¾·ÔÕ·äeÒ\]𝑝:𝑥−𝑎> 0,𝑞:𝑥> 1Iy𝑝L𝑞%ˆ#$ !IzÝ
𝑎%åæu
vtÏ    Ò
Ax{𝑎∣𝑎< 1}
Bx{𝑎∣𝑎≤1}
Cx{𝑎∣𝑎> 1}
Dx{𝑎∣𝑎≥1}
Ó_ 2-1Ï2023·çáèÆ·ÑÖ×ØÒ\]<=𝐴= 𝑥|𝑥2−4 = 0 I𝐵= {𝑥|𝑎𝑥−2 = 0 }Iy𝑥∈𝐴L𝑥∈𝐵
%#$ˆ !IzÝ
𝑎%¬U„‰åæé“%<=tÏ    Ò
Ax{−1,0,1}
Bx{−1,1}
Cx{1}
Dx{−1}
Ó_ 2-2Ï23-24 ‹¾·êëìí·äîÒï𝑥%·ð𝑥2 +𝑎𝑥+ 1 = 0Uk•ˆmñ%Ý
(%$ !
LÏ    Ò
Ax𝑎> 2Ù𝑎< −2
Bx𝑎≥2Ù𝑎≤−2
      
      
Cx𝑎< 1
Dx𝑎> 2
Ó_ 2-3Ï22-23 ‹j·òß·äîÒ\] !𝑝:|𝑥+ 1| > 2I !𝑞:𝑥> 𝑎IÀ¬𝑝L¬𝑞%ˆ#$ 
!Iz𝑎%åæuvLÏ    Ò
Ax𝑎≤1
Bx𝑎≥1
Cx𝑎≥−1
Dx𝑎≤−3
 3  -./01"23/01%45
Ì 3Ï23-24 ‹¾·óíôõ·äîÒjö1e÷ø%LÏ    
Ò
Ax∃𝑥∈𝑅I𝑥≤0
Bx²³U‹•ù
Iœ ˆL=
¤ˆL—
Cx∃𝑥∈{𝑥|𝑥Lú
}I𝑥+ 5Lú
Dx23𝑥∈RIµ…𝑥2 +1 < 2𝑥
Ó_ 3-1Ï2010·ûá·4Òjö1e%51LÏ   Ò
Ax∀𝑥∈𝑅I2𝑥−1 > 0
Bx∀𝑥∈𝑁∗I(𝑥−1)2 > 0
Cx∃𝑥∈𝑅Ilg𝑥< 1
Dx∃𝑥∈𝑅Itan𝑥= 2
Ó_ 3-2Ï23-24 ‹¾·êëêü·ýþÿĀÒjö1L-./01IÀL41%LÏ    Ò
Ax¬U%Ä
Lā
Bx∀𝑥∈𝑅I|𝑥| +1 ≥1
CxU‹•Ý
𝑥Iµ𝑥2 +2𝑥+ 3 = 0
DxU´ĂăÔĄ^Lą^
Ó_ 3-3Ï23-24 ã¾·Ćć·ýþÿĀҐ`jö1
①∀𝑥∈R,𝑥2 +1 > 0q②∀𝑥∈N,𝑥4 ≥1q③∃𝑥∈Z,𝑥3 < 1q④∀𝑥∈𝑄,𝑥2 ≠2x
be41UÏ    Ò
Ax1 •
Bx2 •
Cx3 •
Dx4 •
 4  -./01"23/01%67
Ì 4Ï2024·ÔՓ·ÑÖ×ØÒ1∃𝑥∈[−1,1],𝑥+ |𝑥| < 0%67LÏ    Ò
Ax∃𝑥∈[−1,1],𝑥+ |𝑥| ≥0
Bx∀𝑥∈[−1,1],𝑥+ |𝑥| ≥0
Cx∀𝑥∈(−∞,−1) ∪(1, + ∞),𝑥+ |𝑥| ≥0
Dx∀𝑥∈(−∞,−1) ∪(1, + ∞),𝑥+ |𝑥| < 0
Ó_ 4-1Ï2024·-E·ÑÖ×ØÒ1“∀𝑎> 1IĈ
𝑓(𝑥) = 𝑥𝑎3[𝑎, + ∞)¾VĉĊċ”%67tÏ    Ò
Ax∃𝑎> 1IĈ
𝑓(𝑥) = 𝑥𝑎3[𝑎, + ∞)¾VĉĊČ
Bx∃𝑎> 1IĈ
𝑓(𝑥) = 𝑥𝑎3[𝑎, + ∞)¾ˆVĉĊċ
      
      
Cx∃𝑎≤1IĈ
𝑓(𝑥) = 𝑥𝑎3[𝑎, + ∞)¾VĉĊČ
Dx∃𝑎≤1IĈ
𝑓(𝑥) = 𝑥𝑎3[𝑎, + ∞)¾ˆVĉĊċ
Ó_ 4-2Ï2024 ã·-E·ÿĀÒ\]1 p
∀𝑥> 0Ie𝑥+2𝑥≤4Iz¬𝑝tÏ    Ò
Ax∃𝑥≤0Ie𝑥+2𝑥> 4
Bx∃𝑥> 0Ie𝑥+2𝑥> 4
Cx∃𝑥> 0Ie𝑥+2𝑥≤4
Dx∀𝑥> 0Ie𝑥+2𝑥> 4
Ó_ 4-3Ï2024·Ćí·ÑÖ×ØÒ1“∀𝑥∈0, π
2 Ie𝑥+2sin𝑥> 2𝑥”%67LÏ    Ò
Ax“∀𝑥∈0, π
2 Ie𝑥+2sin𝑥≥2𝑥”
Bx“∀𝑥∈0, π
2 Ie𝑥+2sin𝑥≤2𝑥”
Cx“∃𝑥∈0, π
2 Ie𝑥+2sin𝑥≤2𝑥”
Dx“∃𝑥∈0, π
2 Ie𝑥+2sin𝑥< 2𝑥”
 5  ()1%45+,

Ì 5Ï2023·čĎßďZĐ·ÐÑÒ1“∀𝑥∈[1,2]I𝑥2−𝑎≤0”L41%$ !LÏ    Ò
Ax𝑎> 4
Bx𝑎≥4
Cx𝑎< 1
Dx𝑎≥1
Ó_ 5-1Ï23-24 ‹¾·óíđá·äîÒ\]1𝑝
“∃𝑥∈𝑅I𝑥2−𝑎𝑥+ 3 < 0”t51IzÝ
 a %
åæuvtÏ    Ò
Ax(−∞,−2 3]
Bx(−2 3,2 3)
Cx(−∞,−2 3) ∪(2 3, + ∞)
Dx[−2 3,2 3]
Ó_ 5-2Ï2023·ßàႷÑÖ×ØÒ1“∀𝑥∈[1,2]I𝑥2−𝑎≤0”L41%‹•#$ˆ !L
Ï    Ò
Ax𝑎> 4
Bx𝑎≥4
Cx𝑎< 1
Dx𝑎≥1
Ó_ 5-3Ï23-24 ‹¾·òß·ýþÿĀÒ\]1𝑝:∃𝑥∈[0,1],𝑥2−2𝑥−2 + 𝑎> 0q1𝑞:∀𝑥∈R,𝑥2
−2𝑥−𝑎≠0Iy1𝑝,𝑞žt51IzÝ
𝑎%åæuvtÏ    Ò
Ax[−1,3]
Bx[−1,2]
Cx[0,2]
Dx(−∞,−1]
 6  89:;"<=>=
Ì 6Ï23-24 ‹j·ûáĒē·ýþÿĀÒ\]<=𝐴= {𝑥|−3 < 2𝑥+ 1 < 7}I𝐵= {𝑥|𝑥< −4 Ù𝑥> 2}I
𝐶= {𝑥|3𝑎−2 < 𝑥< 𝑎+ 1}x
(1)+𝐴∩(∁R𝐵)q
(2)y“𝑝:𝑥∈∁R(𝐴∪𝐵)”L“𝑞:𝑥∈𝐶”%ˆ#$ !I+Ý
𝑎%åæuvx
      
      
Ó_ 6-1Ï22-23 ‹¾·ĔáĂĕĆ·ýþÿĀÒ\]<=𝐴= {𝑥∣2 ≤𝑥≤7}I𝐵=
{𝑥∣−3𝑚+ 4 ≤𝑥≤2𝑚−1}IÀ𝐵≠∅x
(1)y𝑝:∀𝑥∈𝐴,𝑥∈𝐵L41I+Ý
𝑚%åæuvq
(2)y𝑞:∃𝑥∈𝐵,𝑥∈𝐴L41I+Ý
𝑚%åæuvx
Ó_ 6-2Ï23-24 ‹¾·Ėė·ýþÿĀÒ\]<=𝐴= {𝑥|𝑚−2 ≤𝑥≤2𝑚+ 1 } 𝐵= {𝑥|−3 ≤𝑥≤5 }.
(1)y𝐴⊆𝐵I+Ý
𝑚%åæuvq
(2)1𝑝
“∃𝑥∈𝐴Iµ…𝑥∈𝐵”L41I+Ý
𝑚%åæuv.
Ó_ 6-3Ï23-24 ‹¾·çáĘę·äîÒ¿<=𝐴= {𝑥|𝑚−3 < 𝑥< 𝑚+ 3,𝑚∈R}I<=𝐵= {𝑥|𝑥< 2Ù
𝑥> 6}x
(1)Ě𝑚= 2£I+𝐴∩𝐵I𝐴∪𝐵q
(2)¿1𝑝:𝑥∈𝐴I1𝑞:𝑥∈𝐵Iy p L q %ˆ#$ !I+Ý
𝑚%åæuvx
‹*V
1xÏ2023·JKsĂ·ÐÑÒy𝑥,𝑦∈𝑅Iz“𝑥> 𝑦”%‹•ˆ#$ !„[LÏ    Ò
Ax|𝑥| > |𝑦|
Bx𝑥2 > 𝑦2
      
      
Cx
𝑥
𝑦> 1
Dx2𝑥−𝑦> 2
2xÏ2024·êëěĜ·‹ÑÒ\]1𝑝:∀𝑥> 1Iln𝑥>
1
3−
1
3𝑥3Iz¬𝑝tÏ    Ò
Ax∀𝑥> 1Iln𝑥≤
1
3−
1
3𝑥3
Bx∃𝑥≤1Iln𝑥<
1
3−
1
3𝑥3
Cx∃𝑥≤1Iln𝑥≤
1
3−
1
3𝑥3
Dx∃𝑥> 1Iln𝑥≤
1
3−
1
3𝑥3
3xÏ2024·ÔÕĝü·ÐÑÒ\]𝑥> 0,𝑦> 0Iz“𝑥+ 𝑦≥1”L“𝑥2 + 𝑦2 ≥1”%Ï    Ò
Axˆ#$ !
Bx#$ˆ !
Cx#$ ! Dx ˆ¤ˆ#
$ !
4xÏ2024·Ćć·ÐÑÒ\]𝑎∈𝑅Iy<=𝑀= {1,𝑎},𝑁= {0,1,2}Iz“𝑎= 0”L“𝑀⊆𝑁”%Ï    Òx
Axˆ#$ !
Bx#$ˆ !
Cx$ !
Dx ˆ¤ˆ#$ !
5xÏ2023·ĔĞ·ÑÖ×ØÒ1𝑝
∀𝑥> 1I 𝑥+2𝑥−3 > 0I1𝑞
∃𝑥∈𝑅I2𝑥2−4𝑥+ 3 = 0IzÏ    Ò
Ax𝑝4𝑞4
Bx𝑝5𝑞5
Cx𝑝5𝑞4
Dx𝑝4𝑞5
6xÏ2023·Mğ·ÑÖ×ØÒ1“∀−2 ≤𝑥≤3,𝑥2−2𝑎≤0”L41%‹•#$ˆ !LÏ    Ò
Ax𝑎≥1
Bx𝑎≥
9
2
Cx𝑎≥5
Dx𝑎≤4
7xÏ2023·ÔÕĝü·‹ÑÒy1“∀𝑥∈𝑅I𝑚≥sin𝑥+ cos𝑥”L41IzÝ
𝑚%åæuvLÏ    Ò
Ax𝑚≥
2
Bx𝑚≥2
Cx𝑚≤−2
Dx𝑚≤−2
8xÏ2024·-E·ÑÖ×ØÒ\]Ġ/
→
𝑎= (4,𝑚)I
→
𝑏= (𝑚−2,2)Iz“𝑚= 4”L“
→
𝑎"
→
𝑏ġĢ”%Ï    Ò
Axˆ#$ !
Bx#$ˆ !
Cx$ !
Dx ˆ¤ˆ#$ !
Ð*ģ
9xÏ23-24 ‹j·çáĤĔ·ĥĦÒjö§¸÷ø%LÏ       Ò.
Ax1“∃𝑥∈RI𝑥+ 1 ≥0”%67L“∀𝑥∈RI𝑥+ 1 < 0”
Bx1“∃𝑥∈RI𝑥2−𝑥+ 1 = 0”L51
Cx“𝑎> 𝑏”L“𝑎2 > 𝑏2”% !
Dx“𝑥> 4”L“𝑥> 2”%ˆ#$ !
10xÏ2024·ÚáħĨĩĪīVĬ·ÑÖ×ØÒ\]<=𝐴= {𝑥|𝑥≤3 }I<=𝐵= {𝑥|𝑥≤𝑚+ 1 }I‰µ𝐴∩𝐵= 𝐴
“”%ˆ#$ !UÏ    Ò
      
      
Ax𝑚> 0
Bx𝑚> 1
Cx𝑚> 3
Dx𝑚> 4
11xÏ23-24 ‹¾·ûáĭĮ·äîÒ\]k•1
Ï1Òy𝑥> 0Iz2𝑥+ 1 > 5qÏ2ÒyÔĄ^tñį
İ^Izɕԡ^%±ıĢmñxzjö§¸÷ø%LÏ    Ò
Ax1Ï2ÒL-./01
Bx1Ï1Ò%67t
23𝑥> 0,2𝑥+ 1 ≤5
Cx1Ï2Ò%67L
23ÔĄ^ˆLñįİ^Iɕԡ^%±ıĢˆmñ
Dx1Ï1ÒsÏ2ÒIJ67ijIL41
ã*Ĵĵ
12xÏ2023·êëěĜ·ÑÖ×ØÒ1𝑝:∃𝑥0 ∈R,𝑥2
0−𝑚𝑥0 +𝑚+ 3 < 0,z1𝑝%67t      
.
13xÏ2023·óííĶ·ÑÖ×ØÒy“𝑥> 2”L“𝑥2−𝑎> 2”%ˆ#$ !Iz a %åæuvL      
x
14xÏ2023·ÔÕá·ÑÖ×ØÒy1“∃𝑥∈𝑅Iµ…𝑥2 +2𝑥−𝑚= 0“””t41IzÝ
𝑚%åæuv
L      
x
Ô*pķ
15xÏ23-24 ‹¾·Ćíĭĸ·äîÒ\]1𝑝:∀𝑥∈𝑅,𝑥2−𝑥−2 > 0x
(1)Ĺ`1𝑝%67q
(2)&'1𝑝%45Iĺ§Æx
16xÏ2023·MğĻü·‹ÑÒ1𝑝
­®𝑥∈𝑅I𝑥2−2𝑚𝑥−3𝑚> 0“”q1𝑞
23𝑥∈𝑅I𝑥2+4𝑚𝑥+ 1 < 0
“”.
(1)y1𝑞t51I+Ý
𝑚%åæuvq
(2)y1𝑝s𝑞UÀÇU‹•t41I+Ý
𝑚%åæuv.
17xÏ23-24 ‹¾·òßļë·äeÒ\]<=𝐴= 𝑥|1
4 ≤2𝑥≤32 I𝐵= 𝑥|𝑥2−4𝑥+ 4−𝑚2 ≤0,𝑚∈R .
      
      
(1)y𝑚= 3I+𝐴∩𝐵q
(2)y23÷Ý
 mIµ…“𝑥∈𝐴”L“𝑥∈𝐵”“”%ˆ#$ !I+÷Ý
 m %åæuv.
18xÏ23-24 ‹¾·ĽľĿŀ·äeÒ\]1𝑝
∃𝑥∈𝑅I𝑎𝑥2 +2𝑥−1 = 0t51x
(1)+Ý
𝑎%åæ<=𝐴q
(2)¿Þĵ<=𝐵= {𝑥|6𝑚−4 < 2𝑥−4 < 2𝑚}Iy“𝑥∈𝐴”L“𝑥∈𝐵”%#$ˆ !I+Ý
𝑚%åæ<=x
19xÏ2024·Łć·ÑÖ×ØÒ¿ XIY t­®<=IłŃ𝑓:𝑋→𝑌x7Ĝ
±­®𝑥1,𝑥2 ∈𝑋Iy𝑥1 ≠𝑥2Iz𝑓(𝑥1)
≠𝑓(𝑥2)I¢£%𝑓tVŃx
(1)Ħ3R→R¾`‹•ÞVŃ%łŃq
(2)ÅÆ
𝑓LVŃ%#$ !L
7­®bc<=𝑍"łŃ𝑔,ℎ:𝑍→𝑋Iy±­®𝑧∈𝑍IU
𝑓(𝑔(𝑧)) = 𝑓(ℎ(𝑧))Iz𝑔= ℎq
(3)ÅÆ
𝑓LVŃ%#$ !L
23łŃ𝜑:𝑌→𝑋Iµ±­®𝑥∈𝑋IU𝜑(𝑓(𝑥)) = 𝑥x