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A
⊕
B
=
(
A
c
⊖
B
s
)
c
Search
Returned 71 matches (100 formulae, 60 docs)
Lookup 2.050 ms, Re-ranking 99.359 ms
Found 20787 tuple postings, 16604 formulae, 6491 documents
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[ documents ]
[ documents-by-formula ]
Doc 1
1.0000
0.0000
11.0000
4.4508
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Mathematical_morphology.html
A
⊕
B
=
(
A
c
⊖
B
s
)
c
A
∙
B
=
(
A
c
∘
B
s
)
c
A
∙
B
=
(
A
c
∘
B
s
)
c
A
∘
B
=
(
A
⊖
B
)
⊕
B
A
∙
B
=
(
A
⊕
B
)
⊖
B
A
⊕
B
=
⋃
b
∈
B
A
b
(
A
⊖
B
)
⊖
C
=
A
⊖
(
B
⊕
C
)
(
A
⊕
B
)
⊆
C
(
A
⊕
B
)
⊕
C
=
A
⊕
(
B
⊕
C
)
Doc 2
0.6923
-2.0000
9.0000
1.3986
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Opening_(morphology).html
A
∙
B
=
(
A
c
∘
B
s
)
c
A
∘
B
=
(
A
⊖
B
)
⊕
B
,
(
A
∘
B
)
∘
B
=
A
∘
B
Doc 3
0.5926
-3.0000
6.0000
1.1481
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Union_(set_theory).html
A
∪
B
=
(
A
C
∩
B
C
)
C
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
Doc 4
0.4615
-8.0000
6.0000
0.4615
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Hit-or-miss_transform.html
A
⊙
B
=
(
A
⊖
C
)
∩
(
A
c
⊖
D
)
Doc 5
0.3871
-5.0000
6.0000
0.5334
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Closing_(morphology).html
A
∙
B
=
(
A
⊕
B
)
⊖
B
,
(
A
∙
B
)
∙
B
=
A
∙
B
Doc 6
0.3871
-5.0000
5.0000
1.0520
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Two-element_Boolean_algebra.html
A
⋅
B
=
A
¯
+
B
¯
¯
A
+
B
=
A
¯
⋅
B
¯
¯
.
A
+
(
B
⋅
C
)
=
(
A
+
B
)
⋅
(
A
+
C
)
.
Doc 7
0.3871
-5.0000
5.0000
0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Context-free_language.html
A
∩
B
=
A
¯
∪
B
¯
¯
Doc 8
0.3871
-7.0000
6.0000
2.0575
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Symmetric_difference.html
A
∪
B
=
(
A
△
B
)
△
(
A
∩
B
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A
△
B
=
(
A
∪
B
)
∖
(
A
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B
)
,
A
△
B
=
(
A
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B
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∪
(
B
∖
A
)
,
A
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(
B
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(
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(
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)
,
A
⊕
B
.
A
⊖
B
.
A
△
B
=
A
c
△
B
c
Doc 9
0.3871
-7.0000
6.0000
1.0669
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Ring_of_sets.html
A
△
B
=
(
A
∖
B
)
∪
(
B
∖
A
)
A
∪
B
=
(
A
△
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)
△
(
A
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B
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A
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B
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A
△
(
A
∩
B
)
.
A
∩
B
=
A
∖
(
A
∖
B
)
.
Doc 10
0.3871
-7.0000
6.0000
0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Jaccard_index.html
A
△
B
=
(
A
∪
B
)
-
(
A
∩
B
)
Doc 11
0.3871
-7.0000
6.0000
0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Linear_logic.html
A
≡
B
=
(
A
⊸
B
)
&
(
B
⊸
A
)
Doc 12
0.3871
-7.0000
4.0000
0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000005/Articles/Generalized_method_of_moments.html
I
-
B
=
(
I
-
B
)
(
I
-
B
)
′
Doc 13
0.3871
-8.0000
6.0000
0.3871
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Set_(mathematics).html
A
Δ
B
=
(
A
∖
B
)
∪
(
B
∖
A
)
.
Doc 14
0.3614
-7.0000
5.0000
0.3614
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/E7_(mathematics).html
A
∘
B
=
(
A
B
+
B
A
)
/
2
Doc 15
0.3614
-9.0000
4.0000
0.3614
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Pfaffian.html
A
1
⊕
A
2
=
[
A
1
0
0
A
2
]
,
Doc 16
0.3614
-20.0000
5.0000
1.1948
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Cartesian_product.html
(
A
×
B
)
c
=
(
A
c
×
B
c
)
∪
(
A
c
×
B
)
∪
(
A
×
B
c
)
.
A
×
(
B
∖
C
)
=
(
A
×
B
)
∖
(
A
×
C
)
,
A
×
(
B
∩
C
)
=
(
A
×
B
)
∩
(
A
×
C
)
,
A
×
(
B
∪
C
)
=
(
A
×
B
)
∪
(
A
×
C
)
,
Doc 17
0.3288
-2.0000
4.0000
0.5595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Schnirelmann_density.html
A
⊕
B
=
𝒩
.
A
⊕
B
Doc 18
0.3288
-2.0000
4.0000
0.3288
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Duality_(mathematics).html
(
A
c
)
c
=
A
Doc 19
0.3288
-3.0000
4.0000
1.7134
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/XOR_swap_algorithm.html
A
⊕
B
=
B
⊕
A
A
⊕
B
A
⊕
B
=
(
A
⊕
A
)
⊕
B
(
A
⊕
B
)
⊕
B
=
A
⊕
(
B
⊕
B
)
(
A
⊕
B
)
⊕
C
=
A
⊕
(
B
⊕
C
)
(
A
⊕
B
)
⊕
A
=
A
⊕
(
A
⊕
B
)
Doc 20
0.3288
-3.0000
4.0000
0.8449
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Complement_(set_theory).html
(
A
c
)
c
=
A
.
(
A
∪
B
)
c
=
A
c
∩
B
c
.
(
A
∩
B
)
c
=
A
c
∪
B
c
.
Doc 21
0.3288
-6.0000
4.0000
0.3288
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Activity_selection_problem.html
B
=
(
A
∖
{
k
}
)
∪
{
1
}
Doc 22
0.3288
-7.0000
4.0000
0.5595
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Dilation_(morphology).html
A
⊕
B
=
⋃
b
∈
B
A
b
.
(
A
⊕
B
)
⊕
C
=
A
⊕
(
B
⊕
C
)
Doc 23
0.3288
-9.0000
4.0000
0.3288
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Erosion_(morphology).html
(
A
⊖
B
)
⊖
C
=
A
⊖
(
B
⊕
C
)
Doc 24
0.2927
-5.0000
6.0000
3.9748
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/De_Morgan's_laws.html
A
c
∪
B
c
⊆
(
A
∩
B
)
c
A
c
∪
B
c
⊆
(
A
∩
B
)
c
A
c
∪
B
c
⊆
(
A
∩
B
)
c
x
∈
(
A
∩
B
)
c
x
∈
(
A
∩
B
)
c
x
∈
(
A
∩
B
)
c
x
∉
(
A
∩
B
)
c
x
∉
(
A
∩
B
)
c
x
∈
(
A
∩
B
)
c
)
(
A
∪
B
)
c
=
A
c
∩
B
c
(
A
∩
B
)
c
=
A
c
∪
B
c
(
A
∩
B
)
c
=
A
c
∪
B
c
(
A
∩
B
)
c
⊆
A
c
∪
B
c
(
A
∩
B
)
c
⊆
A
c
∪
B
c
(
A
∩
B
)
c
⊆
A
c
∪
B
c
Doc 25
0.2778
-9.0000
5.0000
0.2778
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Hadamard_product_(matrices).html
A
∘
(
B
∘
C
)
=
(
A
∘
B
)
∘
C
,
Doc 26
0.2778
-11.0000
5.0000
1.5107
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Algebra_of_sets.html
A
∩
(
B
∪
C
)
=
(
A
∩
B
)
∪
(
A
∩
C
)
A
∪
(
B
∩
C
)
=
(
A
∪
B
)
∩
(
A
∪
C
)
(
A
∪
B
)
C
=
A
C
∩
B
C
(
A
∩
B
)
C
=
A
C
∪
B
C
A
∩
(
A
∪
B
)
=
A
A
∪
(
A
∩
B
)
=
A
A
∩
B
=
A
∖
(
A
∖
B
)
Doc 27
0.2581
-5.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Spherical_trigonometry.html
2
S
=
(
A
+
B
+
C
)
Doc 28
0.2581
-6.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/Computron_tube.html
S
=
(
A
*
B
)
+
C
+
D
Doc 29
0.2581
-6.0000
3.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Q-Vandermonde_identity.html
(
A
+
B
)
m
(
A
+
B
)
n
Doc 30
0.2581
-7.0000
3.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Feynman_parametrization.html
u
=
(
z
-
B
)
/
(
A
-
B
)
Doc 31
0.2581
-8.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Geometric_calculus.html
𝒫
B
(
A
)
=
(
A
⋅
B
-
1
)
B
Doc 32
0.2581
-9.0000
4.0000
0.5161
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Geometric_algebra.html
𝒫
B
(
A
)
=
(
A
⌟
B
-
1
)
⌟
B
𝒫
B
(
A
)
=
(
A
⌟
B
-
1
)
⌟
B
Doc 33
0.2581
-9.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/State-space_representation.html
𝐱
˙
(
t
)
=
(
A
+
B
K
)
𝐱
(
t
)
Doc 34
0.2581
-9.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Amagat's_law.html
B
1
,
2
=
(
B
1
+
B
2
)
/
2
Doc 35
0.2581
-10.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Damping.html
x
(
t
)
=
(
A
+
B
t
)
e
-
ω
0
t
Doc 36
0.2581
-12.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000013/Articles/Multivariate_mutual_information.html
A
~
=
(
A
~
∩
B
~
)
∪
(
A
~
\
B
~
)
Doc 37
0.2581
-12.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/Lamé_function.html
d
2
y
d
x
2
=
(
A
+
B
\weierp
(
x
)
)
y
Doc 38
0.2581
-15.0000
4.0000
0.2581
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000003/Articles/Adder_(electronics).html
C
o
u
t
=
(
A
⋅
B
)
+
(
C
i
n
⋅
(
A
⊕
B
)
)
Doc 39
0.2308
0.0000
3.0000
1.1538
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Exclusive_or.html
A
⊕
B
A
⊕
B
A
⊕
B
A
⊕
B
(
A
⊕
B
)
Doc 40
0.2308
0.0000
3.0000
0.6923
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Direct_sum.html
A
⊕
B
A
⊕
B
A
⊕
B
Doc 41
0.2308
0.0000
3.0000
0.4615
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Logic_gate.html
A
⊕
B
A
⊕
B
¯
Doc 42
0.2308
0.0000
3.0000
0.2308
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000006/Articles/Carry-lookahead_adder.html
A
⊕
B
Doc 43
0.2308
0.0000
3.0000
0.2308
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Direct_sum_of_modules.html
A
⊕
B
Doc 44
0.2308
-6.0000
3.0000
0.3771
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Logical_biconditional.html
A
↔
B
⇔
¬
(
A
⊕
B
)
(
A
↔
B
)
Doc 45
0.1639
-6.0000
5.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Functional_completeness.html
A
∨
B
:=
(
A
→
B
)
→
B
.
Doc 46
0.1639
-6.0000
4.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Reaction_rate.html
A
+
B
⇌
|
A
⋯
B
|
‡
→
P
Doc 47
0.1639
-8.0000
5.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000009/Articles/Monoidal_t-norm_logic.html
A
↔
B
≡
(
A
→
B
)
∧
(
B
→
A
)
Doc 48
0.1639
-8.0000
5.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000010/Articles/T-norm_fuzzy_logics.html
A
↔
B
≡
(
A
→
B
)
∧
(
B
→
A
)
Doc 49
0.1639
-8.0000
5.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000012/Articles/BL_(logic).html
A
↔
B
≡
(
A
→
B
)
∧
(
B
→
A
)
Doc 50
0.1639
-10.0000
5.0000
0.1639
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/XOR_gate.html
A
⋅
B
¯
+
A
¯
⋅
B
≡
(
A
+
B
)
⋅
Doc 51
0.1569
-2.0000
4.0000
0.7738
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/List_of_logic_systems.html
B
→
(
A
∨
B
)
B
→
(
A
∨
B
)
A
→
(
B
→
(
A
∧
B
)
)
A
→
(
B
→
(
A
∧
B
)
)
(
A
↔
B
)
→
(
A
→
B
)
Doc 52
0.1569
-4.0000
4.0000
0.1569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Probability_axioms.html
P
(
B
∖
(
A
∩
B
)
)
Doc 53
0.1569
-5.0000
4.0000
0.1569
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000008/Articles/Crossed_ladders_problem.html
(Eq. 1)
A
B
=
h
(
A
+
B
)
Doc 54
0.1463
-1.0000
3.0000
0.7317
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Logical_conjunction.html
(
A
and
B
)
(
A
and
B
)
(
A
and
B
)
(
A
and
B
)
(
A
and
B
)
Doc 55
0.1463
-1.0000
3.0000
0.1463
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000007/Articles/Tautology_(logic).html
(
A
∧
B
)
Doc 56
0.1463
-1.0000
3.0000
0.1463
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000004/Articles/Comma_category.html
(
A
↓
B
)
Doc 57
0.1463
-2.0000
3.0000
0.1463
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Intuitionistic_logic.html
(
A
∧
B
)
*
Doc 58
0.1463
-5.0000
3.0000
0.2927
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000002/Articles/Combinational_logic.html
A
+
(
A
⋅
B
)
=
A
A
⋅
(
A
+
B
)
=
A
Doc 59
0.1463
-6.0000
3.0000
0.2927
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000014/Articles/Homotopy_type_theory.html
(
A
=
B
)
→
(
A
≃
B
)
(
A
=
B
)
≃
(
A
≃
B
)
Doc 60
0.1463
-9.0000
3.0000
0.2927
testing/NTCIR12_MathIR_WikiCorpus_v2.1.0/MathTagArticles/wpmath0000001/Articles/Natural_deduction.html
A
prop
B
prop
(
A
∧
B
)
prop
∧
F
A
true
B
true
(
A
∧
B
)
true
∧
I