# Logical Tables

### Zeroth Order Logic

Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6
x : 1 1 0 0
y : 1 0 1 0
f0 f0000 0 0 0 0 ( ) false 0
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y
f3 f0011 0 0 1 1 (x) not x ¬x
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y
f5 f0101 0 1 0 1 (y) not y ¬y
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y
f8 f1000 1 0 0 0 x y x and y x ∧ y
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y
f10 f1010 1 0 1 0 y y y
f11 f1011 1 0 1 1 (x (y)) not x without y x → y
f12 f1100 1 1 0 0 x x x
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y
f15 f1111 1 1 1 1 (( )) true 1

### Template Draft

Propositional Forms on Two Variables
L1 L2 L3 L4 L5 L6 Name
x : 1 1 0 0
y : 1 0 1 0
f0 f0000 0 0 0 0 ( ) false 0 Falsity
f1 f0001 0 0 0 1 (x)(y) neither x nor y ¬x ∧ ¬y NNOR
f2 f0010 0 0 1 0 (x) y y and not x ¬x ∧ y Insuccede
f3 f0011 0 0 1 1 (x) not x ¬x Not One
f4 f0100 0 1 0 0 x (y) x and not y x ∧ ¬y Imprecede
f5 f0101 0 1 0 1 (y) not y ¬y Not Two
f6 f0110 0 1 1 0 (x, y) x not equal to y x ≠ y Inequality
f7 f0111 0 1 1 1 (x y) not both x and y ¬x ∨ ¬y NAND
f8 f1000 1 0 0 0 x y x and y x ∧ y Conjunction
f9 f1001 1 0 0 1 ((x, y)) x equal to y x = y Equality
f10 f1010 1 0 1 0 y y y Two
f11 f1011 1 0 1 1 (x (y)) not x without y x → y Implication
f12 f1100 1 1 0 0 x x x One
f13 f1101 1 1 0 1 ((x) y) not y without x x ← y Involution
f14 f1110 1 1 1 0 ((x)(y)) x or y x ∨ y Disjunction
f15 f1111 1 1 1 1 (( )) true 1 Tautology

### Truth Tables

#### Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.

The truth table of NOT p (also written as ~p or ¬p) is as follows:

Logical Negation
p ¬p
F T
T F

The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:

Variant Notations
Notation Vocalization
${\displaystyle {\bar {p}}}$ bar p
${\displaystyle p'\!}$ p prime,

p complement

${\displaystyle !p\!}$ bang p

No matter how it is notated or symbolized, the logical negation ¬p is read as "it is not the case that p", or usually more simply as "not p".

• Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to the initial proposition p. Expressed in symbolic terms, ¬(¬p) ⇔ p.
• Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.

Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as pF, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. But in classical logic, we get a further identity: pq can be defined as ~pq, where ∨ is logical disjunction.

Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic).

#### Logical conjunction

Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.

The truth table of p AND q (also written as p ∧ q, p & q, or p${\displaystyle \cdot }$q) is as follows:

Logical Conjunction
p q p ∧ q
F F F
F T F
T F F
T T T

#### Logical disjunction

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.

The truth table of p OR q (also written as p ∨ q) is as follows:

Logical Disjunction
p q p ∨ q
F F F
F T T
T F T
T T T

#### Logical equality

Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:

Logical Equality
p q p = q
F F T
F T F
T F F
T T T

#### Exclusive disjunction

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.

The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:

Exclusive Disjunction
p q p XOR q
F F F
F T T
T F T
T T F

The following equivalents can then be deduced:

${\displaystyle {\begin{matrix}p+q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\\\&=&(p\lor q)&\land &(\lnot p\lor \lnot q)\\\\&=&(p\lor q)&\land &\lnot (p\land q)\end{matrix}}}$

Generalized or n-ary XOR is true when the number of 1-bits is odd.

#### Logical implication

The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.

The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:

Logical Implication
p q p ⇒ q
F F T
F T T
T F F
T T T

#### Logical NAND

The NAND operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.

The truth table of p NAND q (also written as p | q or p ↑ q) is as follows:

Logical NAND
p q p ↑ q
F F T
F T T
T F T
T T F

#### Logical NOR

The NOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.

The truth table of p NOR q (also written as p ⊥ q or p ↓ q) is as follows:

Logical NOR
p q p ↓ q
F F T
F T F
T F F
T T F

### Exclusive Disjunction

A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)


p + q = (p ∧ !q)  ∨ (!p ∧ B)

= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}

= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}

= (!p ∨ !q) ∧ (p ∨ q)

= !(p ∧ q)  ∧ (p ∨ q)


p + q = (p ∧ ~q)  ∨ (~p ∧ q)

= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)

= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))

= (~p ∨ ~q) ∧ (p ∨ q)

= ~(p ∧ q)  ∧ (p ∨ q)

${\displaystyle {\begin{matrix}p+q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\&=&((p\lor \lnot q)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}$

# Mathematical Symbols

Symbol
Name
Explanation Examples
Category
=
equality x = y means x and y represent the same thing or value. 1 + 1 = 2
is equal to; equals
everywhere

<>

!=
inequation xy means that x and y do not represent the same thing or value. 1 ≠ 2
is not equal to; does not equal
everywhere
<

>

strict inequality x < y means x is less than y.

x > y means x is greater than y.

x ≪ y means x is much less than y.

x ≫ y means x is much greater than y.
3 < 4
5 > 4.

0.003 ≪ 1000000

is less than, is greater than, is much less than, is much greater than
order theory

inequality x ≤ y means x is less than or equal to y.

x ≥ y means x is greater than or equal to y.
3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5
is less than or equal to, is greater than or equal to
order theory
proportionality yx means that y = kx for some constant k. if y = 2x, then yx
is proportional to
everywhere
+
addition 4 + 6 means the sum of 4 and 6. 2 + 7 = 9
plus
arithmetic
disjoint union A1 + A2 means the disjoint union of sets A1 and A2. A1 = {1, 2, 3, 4} ∧ A2 = {2, 4, 5, 7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)}
the disjoint union of ... and ...
set theory
subtraction 9 − 4 means the subtraction of 4 from 9. 8 − 3 = 5
minus
arithmetic
negative sign −3 means the negative of the number 3. −(−5) = 5
negative ; minus
arithmetic
set-theoretic complement A − B means the set that contains all the elements of A that are not in B. {1,2,4} − {1,3,4}  =  {2}
minus; without
set theory
×
multiplication 3 × 4 means the multiplication of 3 by 4. 7 × 8 = 56
times
arithmetic
Cartesian product X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}
the Cartesian product of ... and ...; the direct product of ... and ...
set theory
cross product u × v means the cross product of vectors u and v (1,2,5) × (3,4,−1) =
(−22, 16, − 2)
cross
vector algebra
÷

/
division 6 ÷ 3 or 6/3 means the division of 6 by 3. 2 ÷ 4 = .5

12/4 = 3
divided by
arithmetic
square root x means the positive number whose square is x. √4 = 2
the principal square root of; square root
real numbers
complex square root if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2). √(-1) = i
the complex square root of; square root
complex numbers
| |
absolute value |x| means the distance in the real line (or the complex plane) between x and zero. |3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5
absolute value of
numbers
|
divides A single vertical bar is used to denote divisibility.
a|b means a divides b.
Since 15 = 3×5, it is true that 3|15 and 5|15.
divides
Number Theory
!
factorial n! is the product 1 × 2× ... × n. 4! = 1 × 2 × 3 × 4 = 24
factorial
combinatorics
~
probability distribution X ~ D, means the random variable X has the probability distribution D. X ~ N(0,1), the standard normal distribution
has distribution
statistics

material implication AB means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
x = 2  ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be −2).
implies; if .. then
propositional logic

material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
logical negation The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.
¬(¬A) ⇔ A
x ≠ y  ⇔  ¬(x =  y)
not
propositional logic
logical conjunction or meet in a lattice The statement AB is true if A and B are both true; else it is false. n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.
and
propositional logic, lattice theory
logical disjunction or join in a lattice The statement AB is true if A or B (or both) are true; if both are false, the statement is false. n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
or
propositional logic, lattice theory

exclusive or The statement AB is true when either A or B, but not both, are true. AB means the same. A) ⊕ A is always true, AA is always false.
xor
propositional logic, Boolean algebra
direct sum The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).

Most commonly, for vector spaces U, V, and W, the following consequence is used:
U = VW ⇔ (U = V + W) ∧ (VW = ∅)
direct sum of
Abstract algebra
universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ N: n2 ≥ n.
for all; for any; for each
predicate logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ N: n is even.
there exists
predicate logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ N: n + 5 = 2n.
there exists exactly one
predicate logic
:=

:⇔
definition x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
cosh x := (1/2)(exp x + exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
is defined as
everywhere
{ , }
set brackets {a,b,c} means the set consisting of a, b, and c. N = {0, 1, 2, ...}
the set of ...
set theory
{ : }

{ | }
set builder notation {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. {n ∈ N : n2 < 20} = {0, 1, 2, 3, 4}
the set of ... such that ...
set theory

{}
empty set ∅ means the set with no elements. {} means the same. {n ∈ N : 1 < n2 < 4} = ∅
the empty set
set theory

set membership a ∈ S means a is an element of the set S; a  S means a is not an element of S. (1/2)−1 ∈ N

2−1  N
is an element of; is not an element of
everywhere, set theory

subset (subset) A ⊆ B means every element of A is also element of B.

(proper subset) A ⊂ B means A ⊆ B but A ≠ B.
A ∩ BA; Q ⊂ R
is a subset of
set theory

superset A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B.
A ∪ BB; R ⊃ Q
is a superset of
set theory
set-theoretic union (exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both.
"A or B, but not both".

(inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B.
"A or B or both".
A ⊆ B  ⇔  A ∪ B = B (inclusive)
the union of ... and ...; union
set theory
set-theoretic intersection A ∩ B means the set that contains all those elements that A and B have in common. {x ∈ R : x2 = 1} ∩ N = {1}
intersected with; intersect
set theory
\
set-theoretic complement A \ B means the set that contains all those elements of A that are not in B. {1,2,3,4} \ {3,4,5,6} = {1,2}
minus; without
set theory
( )
function application f(x) means the value of the function f at the element x. If f(x) := x2, then f(3) = 32 = 9.
of
set theory
precedence grouping Perform the operations inside the parentheses first. (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
parentheses
everywhere
f:XY
function arrow fX → Y means the function f maps the set X into the set Y. Let fZ → N be defined by f(x) := x2.
from ... to
set theory
o
function composition fog is the function, such that (fog)(x) = f(g(x)). if f(x) := 2x, and g(x) := x + 3, then (fog)(x) = 2(x + 3).
composed with
set theory

N

natural numbers N means {0, 1, 2, 3, ...}, but see the article on natural numbers for a different convention. {|a| : a ∈ Z} = N
N
numbers

Z

integers Z means {..., −3, −2, −1, 0, 1, 2, 3, ...}. {a, -a : a ∈ N} = Z
Z
numbers

Q

rational numbers Q means {p/q : p,q ∈ Z, q ≠ 0}. 3.14 ∈ Q

π ∉ Q
Q
numbers

R

real numbers R means the set of real numbers. π ∈ R

√(−1) ∉ R
R
numbers

C

complex numbers C means {a + bi : a,b ∈ R}. i = √(−1) ∈ C
C
numbers
arbitrary constant C can be any number, most likely unknown; usually occurs when calculating antiderivatives. if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C
C
integral calculus
infinity ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. limx→0 1/|x| = ∞
infinity
numbers
${\displaystyle \pi }$
pi π is the ratio of a circle's circumference to its diameter. Its value is 3.1415.... A = πr² is the area of a circle with radius r
pi
Euclidean geometry
|| ||
norm ||x|| is the norm of the element x of a normed vector space. ||x+y|| ≤ ||x|| + ||y||
norm of; length of
linear algebra
summation

${\displaystyle \sum _{k=1}^{n}{a_{k}}}$ means a1 + a2 + ... + an.

${\displaystyle \sum _{k=1}^{4}{k^{2}}}$ = 12 + 22 + 32 + 42

= 1 + 4 + 9 + 16 = 30
sum over ... from ... to ... of
arithmetic
product

${\displaystyle \prod _{k=1}^{n}a_{k}}$ means a1a2···an.

${\displaystyle \prod _{k=1}^{4}(k+2)}$ = (1+2)(2+2)(3+2)(4+2)

= 3 × 4 × 5 × 6 = 360
product over ... from ... to ... of
arithmetic
Cartesian product

${\displaystyle \prod _{i=0}^{n}{Y_{i}}}$ means the set of all (n+1)-tuples

(y0,...,yn).

${\displaystyle \prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}}$

the Cartesian product of; the direct product of
set theory
'
derivative f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x. If f(x) := x2, then f '(x) = 2x
... prime; derivative of ...
calculus
indefinite integral or antiderivative ∫ f(x) dx means a function whose derivative is f. x2 dx = x3/3 + C
indefinite integral of ...;; the antiderivative of ...
calculus
definite integral ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. 0b x2  dx = b3/3;
integral from ... to ... of ... with respect to
calculus
gradient f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn). If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
calculus
partial derivative With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. If f(x,y) := x2y, then ∂f/∂x = 2xy
partial derivative of
calculus
boundary M means the boundary of M ∂{x : ||x|| ≤ 2} = {x : ||x|| = 2}
boundary of
topology
perpendicular xy means x is perpendicular to y; or more generally x is orthogonal to y. If lm and mn then l || n.
is perpendicular to
geometry
bottom element x = ⊥ means x is the smallest element. x : x ∧ ⊥ = ⊥
the bottom element
lattice theory
entailment AB means the sentence A entails the sentence B, that is every model in which A is true, B is also true. AA ∨ ¬A
entails
model theory
inference xy means y is derived from x. AB ⊢ ¬B → ¬A
infers or is derived from
propositional logic, predicate logic
normal subgroup NG means that N is a normal subgroup of group G. Z(G) ◅ G
is a normal subgroup of
group theory
/
quotient group G/H means the quotient of group G modulo its subgroup H. {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
mod
group theory
quotient set A/~ means the set of all ~ equivalence classes in A.
set theory
isomorphism GH means that group G is isomorphic to group H Q / {1, −1} ≈ V,
where Q is the quaternion group and V is the Klein four-group.
is isomorphic to
group theory
approximately equal xy means x is approximately equal to y π ≈ 3.14159
is approximately equal to
everywhere
<,>
inner product <x,y> means the inner product between x and y, as defined in an inner product space. The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
<x, y> = 2×-1 + 3×5 = 13
inner product of
vector algebra
tensor product VU means the tensor product of V and U. {1, 2, 3, 4} ⊗ {1,1,2} =
{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
tensor product of
linear algebra

# Work Area

 x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

## Draft 1

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
 0f0 0f1 0 1

 x0 1f0 1f1 1f2 1f3 0 0 1 0 1 1 0 0 1 1

 x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

## Draft 2

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2
 0f0 0f1 0 1

 x0 1f0 1f1 1f2 1f3 0 0 1 0 1 1 0 0 1 1

 x0 x1 2f0 2f1 2f2 2f3 2f4 2f5 2f6 2f7 2f8 2f9 2f10 2f11 2f12 2f13 2f14 2f15 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1