Logical Tables[edit]

Zeroth Order Logic[edit]

**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[edit]

**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

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
${\bar {p}}$	bar p
$p'\!$	p prime, p complement
$!p\!$	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 p → F, 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: p → q can be defined as ~p ∨ q, 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[edit]

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 $\cdot$ q) is as follows:

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

Logical disjunction[edit]

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[edit]

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[edit]

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:

{\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[edit]

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[edit]

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[edit]

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[edit]

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)

{\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}}

Column displays[edit]

Applications

In logic

In mathematics

Other gates

Mathematical Symbols[edit]

Symbol	Name	Explanation	Examples
	Should be read as
	Category
=	equality	x = y means x and y represent the same thing or value.	1 + 1 = 2
	is equal to; equals
	everywhere
≠ <> !=	inequation	x ≠ y 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	y ∝ x means that y = kx for some constant k.	if y = 2x, then y ∝ x
	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	A ⇒ B 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 A ∧ B 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 A ∨ B 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 A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.	(¬A) ⊕ A is always true, A ⊕ A 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 = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = ∅)
	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
∈ $\notin$	set membership	a ∈ S means a is an element of the set S; a $\notin$ S means a is not an element of S.	(1/2)⁻¹ ∈ N 2⁻¹ $\notin$ 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 ∩ B ⊆ A; 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 ∪ B ⊇ B; 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:X→Y	function arrow	f: X → Y means the function f maps the set X into the set Y.	Let f: Z → 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 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 ℤ	integers	Z means {..., −3, −2, −1, 0, 1, 2, 3, ...}.	{a, -a : a ∈ N} = Z
	Z
	numbers
Q ℚ	rational numbers	Q means {p/q : p,q ∈ Z, q ≠ 0}.	3.14 ∈ Q π ∉ Q
	Q
	numbers
R ℝ	real numbers	R means the set of real numbers.	π ∈ R √(−1) ∉ R
	R
	numbers
C 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.	lim_x→0 1/\|x\| = ∞
	infinity
	numbers
$\pi$	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	$\sum _{k=1}^{n}{a_{k}}$ means a₁ + a₂ + ... + a_n.	$\sum _{k=1}^{4}{k^{2}}$ = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
	sum over ... from ... to ... of
	arithmetic
∏	product	$\prod _{k=1}^{n}a_{k}$ means a₁a₂···a_n.	$\prod _{k=1}^{4}(k+2)$ = (1+2)(2+2)(3+2)(4+2) = 3 × 4 × 5 × 6 = 360
	product over ... from ... to ... of
	arithmetic
	Cartesian product	$\prod _{i=0}^{n}{Y_{i}}$ means the set of all (n+1)-tuples (y₀,...,y_n).	$\prod _{n=1}^{3}{\mathbb {R} }=\mathbb {R} \times \mathbb {R} \times \mathbb {R} =\mathbb {R} ^{3}$
	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	∫_a^b f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.	∫₀^b x² dx = b³/3;
	integral from ... to ... of ... with respect to
	calculus
∇	gradient	∇f (x₁, …, x_n) is the vector of partial derivatives (df / dx₁, …, df / dx_n).	If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
	del, nabla, gradient of
	calculus
∂	partial derivative	With f (x₁, …, x_n), ∂f/∂x_i is the derivative of f with respect to x_i, 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	x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y.	If l⊥m and m⊥n then l \|\| n.
	is perpendicular to
	geometry
	bottom element	x = ⊥ means x is the smallest element.	∀x : x ∧ ⊥ = ⊥
	the bottom element
	lattice theory
⊧	entailment	A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true.	A ⊧ A ∨ ¬A
	entails
	model theory
⊢	inference	x ⊢ y means y is derived from x.	A → B ⊢ ¬B → ¬A
	infers or is derived from
	propositional logic, predicate logic
◅	normal subgroup	N ◅ G 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	G ≈ H 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	x ≈ y 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	V ⊗ U 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[edit]

Binary Operations
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[edit]

TRUTH TABLES FOR THE BOOLEAN OPERATIONS OF ARITY UP TO 2

Constants
	⁰f₀	⁰f₁
	0	1

Unary Operations
x₀	¹f₀	¹f₁	¹f₂	¹f₃
0	0	1	0	1
1	0	0	1	1

Binary Operations
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[edit]