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Truth table

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Mathematical table used in logic

Logical connectives

NOT	$\neg A,-A,{\overline {A}},{\sim }A$
AND	$A\land B,A\cdot B,AB,A\mathop {\&} B,A\mathop {\&\&} B$
NAND	$A\mathrel {\overline {\land }} B,A\uparrow B,A\mid B,{\overline {A\cdot B}}$
OR	$A\lor B,A+B,A\mid B,A\parallel B$
NOR	$A\mathrel {\overline {\lor }} B,A\downarrow B,{\overline {A+B}}$
XNOR	$A\odot B,{\overline {A\mathrel {\overline {\lor }} B}}$
└equivalent	$A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B$
XOR	$A\mathrel {\underline {\lor }} B,A\oplus B$
└ nonequivalent	$A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B$
implies	$A\Rightarrow B,A\supset B,A\rightarrow B$
nonimplication (NIMPLY)	$A\not \Rightarrow B,A\not \supset B,A\nrightarrow B$
converse	$A\Leftarrow B,A\subset B,A\leftarrow B$
converse nonimplication	$A\not \Leftarrow B,A\not \subset B,A\nleftarrow B$

Related concepts

Applications

History

[edit]

Irving Anellis's research shows thatC.S. Peirce appears to be the earliest logician (in 1883) to devise a truth table matrix.^[4]

From the summary of Anellis's paper:^[4]

In 1997, John Shosky discovered, on theverso of a page of the typed transcript ofBertrand Russell's 1912 lecture on "The Philosophy of Logical Atomism" truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883–84 in connection with the composition of Peirce's "On the Algebra of Logic: A Contribution to the Philosophy of Notation" that appeared in theAmerican Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional.

Applications

[edit]

Truth tables can be used to prove many otherlogical equivalences. For example, consider the following truth table:

$(p\rightarrow q)\equiv (\neg p\vee q)$
$p {\displaystyle p}$	$q {\displaystyle q}$	$\neg p$	$\neg p\vee q$	$p\rightarrow q$
T	T	F	T	T
T	F	F	F	F
F	T	T	T	T
F	F	T	T	T

This demonstrates the fact that $p\rightarrow q$ islogically equivalent to $\neg p\vee q$ .

Truth table for logic gates

[edit]

Here is a truth table that gives definitions of each of the 6 possible 2-inputlogic gate functions of two Boolean variables P and Q:

$P {\displaystyle P}$	$Q {\displaystyle Q}$	$P\land Q$	$P\vee Q$	$P\uparrow Q$	$P\downarrow Q$	$P\nleftrightarrow Q$	$P\leftrightarrow Q$
T	T	T	T	F	F	F	T
T	F	F	T	T	F	T	F
F	T	F	T	T	F	T	F
F	F	F	F	T	T	F	T
Name (function)		AND (conjunction)	OR (disjunction)	NAND (non-conjunction)	NOR (non-disjunction)	XOR (non-equivalence)	XNOR (equivalence)
where T meanstrueand F meansfalse

Condensed truth tables for binary operators

[edit]

For binary operators, a condensed form of truth table is also used, where the row headings and the column headings specify the operands and the table cells specify the result. For example,Boolean logic uses this condensed truth table notation:

∧	T	F
T	T	F
F	F	F

∨	T	F
T	T	T
F	T	F

This notation is useful especially if the operations are commutative, although one can additionally specify that the rows are the first operand and the columns are the second operand. This condensed notation is particularly useful in discussing multi-valued extensions of logic, as it significantly cuts down on combinatoric explosion of the number of rows otherwise needed. It also provides for quickly recognizable characteristic "shape" of the distribution of the values in the table which can assist the reader in grasping the rules more quickly.

Truth tables in digital logic

[edit]

Truth tables are also used to specify the function ofhardware look-up tables (LUTs) indigital logic circuitry. For an n-input LUT, the truth table will have⁠ $2^{n}$ ⁠ values (or rows in the above tabular format), completely specifying a Boolean function for the LUT. By representing each Boolean value as abit in abinary number, truth table values can be efficiently encoded asinteger values inelectronic design automation (EDA)software. For example, a 32-bit integer can encode the truth table for a LUT with up to 5 inputs.

When using an integer representation of a truth table, the output value of the LUT can be obtained by calculating a bit indexk based on the input values of the LUT, in which case the LUT's output value is thekth bit of the integer. For example, to evaluate the output value of a LUT given anarray ofn Boolean input values, the bit index of the truth table's output value can be computed as follows: if theith input is true, let $V_{i}=1$ , else let $V_{i}=0$ . Then thekth bit of the binary representation of the truth table is the LUT's output value, where $k=V_{0}\times 2^{0}+V_{1}\times 2^{1}+V_{2}\times 2^{2}+\dots +V_{n-1}\times 2^{n-1}.$

Truth tables are a simple and straightforward way to encode Boolean functions, however given theexponential growth in size as the number of inputs increase, they are not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations andbinary decision diagrams.

Applications of truth tables in digital electronics

[edit]

In digital electronics and computer science (fields of applied logic engineering and mathematics), truth tables can be used to reduce basic Boolean operations to simple correlations of inputs to outputs, without the use oflogic gates or code. For example, a binary addition can be represented with the truth table:

Binary addition
A	B	C	R
T	T	T	F
T	F	F	T
F	T	F	T
F	F	F	F

where A is the first operand, B is the second operand, C is the carry digit, and R is the result.

This truth table is read left to right:

Value pair (A, B) equals value pair (C, R).
Or for this example, A plus B equal result R, with the Carry C.

This table does not describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.

With respect to the result, this example may be arithmetically viewed as modulo 2 binary addition, and as logically equivalent to the exclusive-or (exclusive disjunction) binary logic operation.

In this case it can be used for only very simple inputs and outputs, such as 1s and 0s. However, if the number of types of values one can have on the inputs increases, the size of the truth table will increase.

For instance, in an addition operation, one needs two operands, A and B. Each can have one of two values, zero or one. The number of combinations of these two values is 2 × 2, or four. So the result is four possible outputs of C and R. If one were to use base 3, the size would increase to 3 × 3, or nine possible outputs.

The first "addition" example above is called a half-adder. A full-adder is when the carry from the previous operation is provided as input to the next adder. Thus, a truth table of eight rows would be needed to describe afull adder's logic:

A B C* | C R0 0 0  | 0 00 1 0  | 0 11 0 0  | 0 11 1 0  | 1 00 0 1  | 0 10 1 1  | 1 01 0 1  | 1 01 1 1  | 1 1Same as previous, but..C* = Carry from previous adder

Methods of writing truth tables

[edit]

Regarding theguide columns^[5] to the left of a table, which representpropositional variables, different authors have different recommendations about how to fill them in, although this is of no logical significance.^[6]

Alternating method

[edit]

Lee Archie, a professor atLander University, recommends this procedure, which is commonly followed in published truth-tables:

Write out the number of variables (corresponding to the number of statements) in alphabetical order.
The number of lines needed is 2ⁿ where n is the number of variables. (E. g., with three variables, 2³ = 8).
Start in the right-hand column and alternateT's andF's until you run out of lines.
Then move left to the next column and alternate pairs ofT's andF's until you run out of lines.
Then continue to the next left-hand column and double the numbers ofT's andF's until completed.^[5]

This method results in truth-tables such as the following table forP → (Q ∨R → (R → ¬P)), produced byStephen Cole Kleene:^[7]

$P {\displaystyle P}$	$Q {\displaystyle Q}$	$R {\displaystyle R}$	$P\rightarrow (Q\vee R\rightarrow (R\rightarrow \neg P))$
T	T	T	F
T	T	F	T
T	F	T	F
T	F	F	T
F	T	T	T
F	T	F	T
F	F	T	T
F	F	F	T

Combinatorial method

[edit]

Colin Howson, on the other hand, believes that "it is a good practical rule" to do the following:

to start with all Ts, then all the ways (three) two Ts can be combined with one F, then all the ways (three) one T can be combined with two Fs, and then finish with all Fs. If a compound is built up from n distinct sentence letters, its truth table will have 2ⁿ rows, since there are two ways of assigning T or F to the first letter, and for each of these there will be two ways of assigning T or F to the second, and for each of these there will be two ways of assigning T or F to the third, and so on, giving 2.2.2. …, n times, which is equal to 2ⁿ.^[6]

This results in truth tables like this table "showing that(A→C)∧(B→C) and(A∨B)→C aretruth-functionally equivalent", modeled after a table produced byHowson:^[6]

$A {\displaystyle A}$	$B {\displaystyle B}$	$C {\displaystyle C}$	$(A\rightarrow C)\land (B\rightarrow C)$	$(A\vee B)\rightarrow C$
T	T	T	T	T
T	T	F	F	F
T	F	T	T	T
F	T	T	T	T
F	F	T	T	T
F	T	F	F	F
T	F	F	F	F
F	F	F	T	T

Size of truth tables

[edit]

If there aren input variables then there are 2ⁿ possible combinations of their truth values. A given function may produce true or false for each combination so the number of different functions ofn variables is thedouble exponential 2^2ⁿ.

n	2ⁿ	2^2ⁿ
0	1	2
1	2	4
2	4	16
3	8	256
4	16	65,536
5	32	4,294,967,296	≈ 4.3×10⁹
6	64	18,446,744,073,709,551,616	≈ 1.8×10¹⁹
7	128	340,282,366,920,938,463,463,374,607,431,768,211,456	≈ 3.4×10³⁸
8	256	115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936	≈ 1.2×10⁷⁷

Truth tables for functions of three or more variables are rarely given.

Function Tables

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It can be useful to have the output of a truth table expressed as a function of some variable values, instead of just a literal truth or false value. These may be called "function tables" to differentiate them from the more general "truth tables".^[8] For example, one value,G, may be used with an XOR gate to conditionally invert another value,X. In other words, whenG is false, the output isX, and whenG is true, the output is ${\textstyle \neg X}$ . The function table for this would look like:

$G {\displaystyle G}$	$G\nleftrightarrow X$
F	$X {\displaystyle X}$
T	$\neg X$

Similarly, a 4-to-1multiplexer with select inputs $S_{0}$ and $S_{1}$ , data inputsA,B,C andD, and outputZ (as displayed in the image) would have this function table:

$S_{1}$	$S_{0}$	Z
F	F	A
F	T	B
T	F	C
T	T	D

Sentential operator truth tables

[edit]

Overview table

[edit]

Here is an extended truth table giving definitions of all sixteen possible truth functions of two Boolean variablesp andq:^{[note 1]}

$p {\displaystyle p}$	$q {\displaystyle q}$	$\bot$	$p\downarrow q$	$p\nleftarrow q$	$\neg p$	$p\nrightarrow q$	$\neg q$	$p\nleftrightarrow q$	$p\uparrow q$	$p\land q$	$p\leftrightarrow q$	$q {\displaystyle q}$	$p\rightarrow q$	$p {\displaystyle p}$	$p\leftarrow q$	$p\vee q$	$\top$
T	T	F	F	F	F	F	F	F	F	T	T	T	T	T	T	T	T
T	F	F	F	F	F	T	T	T	T	F	F	F	F	T	T	T	T
F	T	F	F	T	T	F	F	T	T	F	F	T	T	F	F	T	T
F	F	F	T	F	T	F	T	F	T	F	T	F	T	F	T	F	T

Com
Assoc
Adj		$\bot$	$p\downarrow q$	$p\nrightarrow q$	$\neg q$	$p\nleftarrow q$	$\neg p$	$p\nleftrightarrow q$	$p\uparrow q$	$p\land q$	$p\leftrightarrow q$	$p {\displaystyle p}$	$p\leftarrow q$	$q {\displaystyle q}$	$p\rightarrow q$	$p\vee q$	$\top$
Neg		$\top$	$p\vee q$	$p\leftarrow q$	$p {\displaystyle p}$	$p\rightarrow q$	$q {\displaystyle q}$	$p\leftrightarrow q$	$p\land q$	$p\uparrow q$	$p\nleftrightarrow q$	$\neg q$	$p\nrightarrow q$	$\neg p$	$p\nleftarrow q$	$p\downarrow q$	$\bot$
Dual		$\top$	$p\uparrow q$	$p\rightarrow q$	$\neg p$	$p\leftarrow q$	$\neg q$	$p\leftrightarrow q$	$p\downarrow q$	$p\vee q$	$p\nleftrightarrow q$	$q {\displaystyle q}$	$p\nleftarrow q$	$p {\displaystyle p}$	$p\nrightarrow q$	$p\land q$	$\bot$
L id				F				F		T	T	T, F	T			F
R id						F		F		T	T			T, F	T	F

where

T = true.

F = false.

TheCom row indicates whether an operator,op, iscommutative –P opQ =Q opP.

TheAssoc row indicates whether an operator,op, isassociative –(P opQ) opR =P op (Q opR).

TheAdj row shows the operatorop2 such thatP opQ =Q op2P.

TheNeg row shows the operatorop2 such thatP opQ = ¬(P op2Q).

TheDual row shows thedual operation obtained by interchanging T with F, and AND with OR.

TheL id row shows the operator'sleft identities if it has any valuesI such thatI opQ =Q.

TheR id row shows the operator'sright identities if it has any valuesI such thatP opI =P.^{[note 2]}

Wittgenstein table

[edit]

In proposition 5.101 of theTractatus Logico-Philosophicus,^[9]Wittgenstein listed the table above as follows:

	Truthvalues		Operator		Operation name	Tractatus^{[note 3]}
0	(F F F F)(p, q)	⊥	false	Opq	Contradiction	p and not p; and q and not q
1	(F F F T)(p, q)	NOR	p ↓q	Xpq	Logical NOR	neitherp norq
2	(F F T F)(p, q)	↚	p ↚q	Mpq	Converse nonimplication	q and notp
3	(F F T T)(p, q)	¬p,~p	¬p	Np,Fpq	Negation	notp
4	(F T F F)(p, q)	↛	p ↛q	Lpq	Material nonimplication	p and notq
5	(F T F T)(p, q)	¬q,~q	¬q	Nq,Gpq	Negation	notq
6	(F T T F)(p, q)	XOR	p ⊕q	Jpq	Exclusive disjunction	p orq, but not both
7	(F T T T)(p, q)	NAND	p ↑q	Dpq	Logical NAND	not bothp andq
8	(T F F F)(p, q)	AND	p ∧q	Kpq	Logical conjunction	p andq
9	(T F F T)(p, q)	XNOR	piffq	Epq	Logical biconditional	ifp thenq; and ifq thenp
10	(T F T F)(p, q)	q	q	Hpq	Projection function	q
11	(T F T T)(p, q)	p →q	ifp thenq	Cpq	Material implication	ifp thenq
12	(T T F F)(p, q)	p	p	Ipq	Projection function	p
13	(T T F T)(p, q)	p ←q	ifq thenp	Bpq	Converse implication	ifq thenp
14	(T T T F)(p, q)	OR	p ∨q	Apq	Logical disjunction	p orq
15	(T T T T)(p, q)	⊤	true	Vpq	Tautology	if p then p; and if q then q

The truth table represented by each row is obtained by appending the sequence given inTruthvalues_row to the table^{[note 3]}

p	T	T	F	F
q	T	F	T	F

For example, the table

p	T	T	F	F
q	T	F	T	F
11	T	F	T	T

represents the truth table forMaterial implication. Logical operators can also be visualized usingVenn diagrams.

Nullary operations

[edit]

There are 2 nullary operations:

Always true
Never true, unaryfalsum

Logical true

[edit]

The output value is always true, because this operator has zero operands and therefore no input values

p	T
T	T
F	T

Logical false

[edit]

The output value is never true: that is, always false, because this operator has zero operands and therefore no input values

p	F
T	F
F	F

Unary operations

[edit]

There are 2 unary operations:

Unaryidentity
Unarynegation

Logical identity

[edit]

Logical identity is anoperation on onelogical value p, for which the output value remains p.

The truth table for the logical identity operator is as follows:

p	p
T	T
F	F

Logical negation

[edit]

Logical negation is anoperation on onelogical value, typically the value of aproposition, that produces a value oftrue if its operand is false and a value offalse if its operand is true.

The truth table forNOT p (also written as¬p,Np,Fpq, or~p) is as follows:

p	¬p
T	F
F	T

Binary operations

[edit]

There are 16 possibletruth functions of twobinary variables, each operator has its own name.

Logical conjunction (AND)

[edit]

Logical conjunction is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if both of its operands are true.

The truth table forp AND q (also written asp ∧ q,Kpq,p & q, orp $\cdot$ q) is as follows:

p	q	p ∧q
T	T	T
T	F	F
F	T	F
F	F	F

In ordinary language terms, if bothp andq are true, then the conjunctionp ∧q is true. For all other assignments of logical values top and toq the conjunctionp ∧ q is false.

It can also be said that ifp, thenp ∧q isq, otherwisep ∧q isp.

Logical disjunction (OR)

[edit]

Logical disjunction is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if at least one of its operands is true.

The truth table forp OR q (also written asp ∨ q,Apq,p || q, orp + q) is as follows:

p	q	p ∨q
T	T	T
T	F	T
F	T	T
F	F	F

Stated in English, ifp, thenp ∨q isp, otherwisep ∨q isq.

Logical implication

[edit]

Logical implication and thematerial conditional are both associated with anoperation on twological values, typically the values of twopropositions, which produces a value offalse if the first operand is true and the second operand is false, and a value oftrue otherwise.

The truth table associated with the logical implicationp implies q (symbolized asp ⇒ q, or more rarelyCpq) is as follows:

p	q	p ⇒q
T	T	T
T	F	F
F	T	T
F	F	T

The truth table associated with the material conditionalif p then q (symbolized asp → q) is as follows:

p	q	p →q
T	T	T
T	F	F
F	T	T
F	F	T

p ⇒ q andp → q are equivalent to¬p ∨ q.

Logical equality

[edit]

Logical equality (also known asbiconditional orexclusive nor) is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if both operands are false or both operands are true.

The truth table forp XNOR q (also written asp ↔ q,Epq,p = q, orp ≡ q) is as follows:

p	q	p ↔q
T	T	T
T	F	F
F	T	F
F	F	T

So p EQ q is true if p and q have the sametruth value (both true or both false), and false if they have different truth values.

Exclusive disjunction

[edit]

Exclusive disjunction is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if one but not both of its operands is true.

The truth table forp XOR q (also written asJpq, orp ⊕ q) is as follows:

p	q	p ⊕q
T	T	F
T	F	T
F	T	T
F	F	F

For two propositions,XOR can also be written as (p ∧ ¬q) ∨ (¬p ∧ q).

Logical NAND

[edit]

Thelogical NAND is anoperation on twological values, typically the values of twopropositions, that produces a value offalse if both of its operands are true. In other words, it produces a value oftrue if at least one of its operands is false.

The truth table forp NAND q (also written asp ↑ q,Dpq, orp | q) is as follows:

p	q	p ↑q
T	T	F
T	F	T
F	T	T
F	F	T

It is frequently useful to express a logical operation as acompound operation, that is, as an operation that is built up or composed from other operations. Many such compositions are possible, depending on the operations that are taken as basic or "primitive" and the operations that are taken as composite or "derivative".

In the case of logical NAND, it is clearly expressible as a compound of NOT and AND.

The negation of a conjunction: ¬(p ∧ q), and the disjunction of negations: (¬p) ∨ (¬q) can be tabulated as follows:

p	q	p ∧ q	¬(p ∧ q)	¬p	¬q	(¬p) ∨ (¬q)
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

Logical NOR

[edit]

Thelogical NOR is anoperation on twological values, typically the values of twopropositions, that produces a value oftrue if both of its operands are false. In other words, it produces a value offalse if at least one of its operands is true. ↓ is also known as thePeirce arrow after its inventor,Charles Sanders Peirce, and is aSole sufficient operator.

The truth table forp NOR q (also written asp ↓ q, orXpq) is as follows:

p	q	p ↓q
T	T	F
T	F	F
F	T	F
F	F	T

The negation of a disjunction ¬(p ∨ q), and the conjunction of negations (¬p) ∧ (¬q) can be tabulated as follows:

p	q	p ∨ q	¬(p ∨ q)	¬p	¬q	(¬p) ∧ (¬q)
T	T	T	F	F	F	F
T	F	T	F	F	T	F
F	T	T	F	T	F	F
F	F	F	T	T	T	T

Inspection of the tabular derivations for NAND and NOR, under each assignment of logical values to the functional argumentsp andq, produces the identical patterns of functional values for ¬(p ∧ q) as for (¬p) ∨ (¬q), and for ¬(p ∨ q) as for (¬p) ∧ (¬q). Thus the first and second expressions in each pair are logically equivalent, and may be substituted for each other in all contexts that pertain solely to their logical values.

This equivalence is one ofDe Morgan's laws.

Notes

[edit]

^Information about notation may be found in (Bocheński 1959), (Enderton 2001), and (Quine 1982).
^The operators here with equal left and right identities (XOR, AND, XNOR, and OR) are alsocommutative monoids because they are alsoassociative. While this distinction may be irrelevant in a simple discussion of logic, it can be quite important in more advanced mathematics. For example, incategory theory anenriched category is described as a basecategory enriched over a monoid, and any of these operators can be used for enrichment.
^^a ^bWittgenstein used a different mapping. In proposition 5.101 of the Tractatus one has to appendTruthvalues_row to the table
p T F T F
q T T F F
This explains whyTractatus_row in the table given here does not point to the sameTruthvalues_row as in the Tractatus.

p	T	F	T	F
q	T	T	F	F

References

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^Enderton 2001
^von Wright, Georg Henrik (1955). "Ludwig Wittgenstein, A Biographical Sketch".The Philosophical Review.64 (4): 527–545 (p. 532, note 9).doi:10.2307/2182631.JSTOR 2182631.
^Post, Emil (July 1921). "Introduction to a general theory of elementary propositions".American Journal of Mathematics.43 (3):163–185.doi:10.2307/2370324.hdl:2027/uiuo.ark:/13960/t9j450f7q.JSTOR 2370324.
^^a ^bAnellis, Irving H. (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table".History and Philosophy of Logic.33:87–97.doi:10.1080/01445340.2011.621702.S2CID 170654885.
^^a ^b"How to Construct a Truth Table".philosophy.lander.edu. Retrieved2024-04-05.
^^a ^b ^cHowson, Colin (1997).Logic with trees: an introduction to symbolic logic. London; New York: Routledge. p. 10.ISBN 978-0-415-13342-5.
^Kleene, Stephen Cole (2013).Mathematical Logic. Dover Books on Mathematics. Courier Corporation. p. 11.ISBN 9780486317076.
^Mano, M. Morris; Ciletti, Michael (2018-07-13).Digital Design, Global Edition (6th ed.). Pearson Education, Limited.ISBN 9781292231167.
^Wittgenstein, Ludwig (1922).Tractatus Logico-Philosophicus(PDF). Proposition 5.101.

Works cited

[edit]

Bocheński, Józef Maria (1959).A Précis of Mathematical Logic. Translated by Bird, Otto. D. Reidel.doi:10.1007/978-94-017-0592-9.ISBN 978-94-017-0592-9.{{cite book}}:ISBN / Date incompatibility (help)
Enderton, H. (2001).A Mathematical Introduction to Logic (2nd ed.). Harcourt Academic Press.ISBN 0-12-238452-0.
Quine, W.V. (1982).Methods of Logic (4th ed.). Harvard University Press.ISBN 978-0-674-57175-4.

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Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps, cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive

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v t e Diagrams in logic
Venn diagram Argument map Proof net Square of opposition Porphyrian tree Karnaugh map Binary decision diagram Propositional directed acyclic graph Sentential decision diagram Truth table Sequent calculus Method of analytic tableaux Existential graph Euler diagram Carroll diagram Randolph diagram