truth table formulas

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables (Enderton, 2001). In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.

A truth table has one column for each input variable (for example, P and Q), and one final column showing all of the possible results of the logical operation that the table represents (for example, P XOR Q). Each row of the truth table contains one possible configuration of the input variables (for instance, P=true Q=false), and the result of the operation for those values. See the examples below for further clarification. Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published in 1921.[1] Such a system was also independently proposed in 1921 by Emil Leon Post.[2] An even earlier iteration of the truth table has also been found in unpublished manuscripts by Charles Sanders Peirce from 1893, antedating both publications by nearly 30 years.[3]

Unary operations

There are 4 unary operations:

Always true

Never true, unary falsum

Unary Identity

Unary negation

Logical true

The output value is always true, regardless of the input value of p

Logical True

p T

T T

F T

Logical false

The output value is never true: that is, always false, regardless of the input value of p

Logical False

p F

T F

F F

Logical identity

Logical identity is an operation on one logical value p, for which the output value remains p.

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

Logical Identity

p p

T T

F F

Logical negation

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

The truth table for NOT p (also written as ¬p, Np, Fpq, or ~p) is as follows:

Logical Negation

p ¬p

T F

F T

Binary operations

There are 16 possible truth functions of two binary variables:

Truth table for all binary logical operators

Here is an extended truth table giving definitions of all possible truth functions of two Boolean variables P and Q:[note 1]

p q F0 NOR1 ↚2 ¬p3 ↛4 ¬q5 XOR6 NAND7 AND8 XNOR9 q10 →11 p12 ←13 OR14 T15

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 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

Adj F0 NOR1 ↛4 ¬q5 ↚2 ¬p3 XOR6 NAND7 AND8 XNOR9 p12 ←13 q10 →11 OR14 T15

Neg T15 OR14 ←13 p12 →11 q10 XNOR9 AND8 NAND7 XOR6 ¬q5 ↛4 ¬p3 ↚2 NOR1 F0

Dual T15 NAND7 →11 ¬p3 ←13 ¬q5 XNOR9 NOR1 OR14 XOR6 q10 ↚2 p12 ↛4 AND8 F0

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.

The Com row indicates whether an operator, op, is commutative - P op Q = Q op P.

The Adj row shows the operator op2 such that P op Q = Q op2 P

The Neg row shows the operator op2 such that P op Q = ¬(Q op2 P)

The Dual row shows the dual operation obtained by interchanging T with F, and AND with OR.

The L id row shows the operator's left identities if it has any - values I such that I op Q = Q.

The R id row shows the operator's right identities if it has any - values I such that P op I = P.[note 2]

The four combinations of input values for p, q, are read by row from the table above. The output function for each p, q combination, can be read, by row, from the table.

Key:

The following table is oriented by column, rather than by row. There are four columns rather than four rows, to display the four combinations of p, q, as input.

p: T T F F

q: T F T F

There are 16 rows in this key, one row for each binary function of the two binary variables, p, q. For example, in row 2 of this Key, the value of Converse nonimplication ('{\displaystyle \nleftarrow }\nleftarrow ') is solely T, for the column denoted by the unique combination p=F, q=T; while in row 2, the value of that '{\displaystyle \nleftarrow }\nleftarrow ' operation is F for the three remaining columns of p, q. The output row for {\displaystyle \nleftarrow }\nleftarrow  is thus

2: F F T F

and the 16-row[4] key is

[4] operator Operation name

0 (F F F F)(p, q) ⊥ false, Opq Contradiction

1 (F F F T)(p, q) NOR p ↓ q, Xpq Logical NOR

2 (F F T F)(p, q) ↚ p ↚ q, Mpq Converse nonimplication

3 (F F T T)(p, q) ¬p, ~p ¬p, Np, Fpq Negation

4 (F T F F)(p, q) ↛ p ↛ q, Lpq Material nonimplication

5 (F T F T)(p, q) ¬q, ~q ¬q, Nq, Gpq Negation

6 (F T T F)(p, q) XOR p ⊕ q, Jpq Exclusive disjunction

7 (F T T T)(p, q) NAND p ↑ q, Dpq Logical NAND

8 (T F F F)(p, q) AND p ∧ q, Kpq Logical conjunction

9 (T F F T)(p, q) XNOR p If and only if q, Epq Logical biconditional

10 (T F T F)(p, q) q q, Hpq Projection function

11 (T F T T)(p, q) p → q if p then q, Cpq Material implication

12 (T T F F)(p, q) p p, Ipq Projection function

13 (T T F T)(p, q) p ← q p if q, Bpq Converse implication

14 (T T T F)(p, q) OR p ∨ q, Apq Logical disjunction

15 (T T T T)(p, q) ⊤ true, Vpq Tautology

Logical operators can also be visualized using Venn diagrams.

Logical conjunction (AND)

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

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

Logical conjunction

p q p ∧ q

T T T

T F F

F T F

F F F

In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values to p and to q the conjunction p ∧ q is false.

It can also be said that if p, then p ∧ q is q, otherwise p ∧ q is p.

Logical disjunction (OR)

Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if at least one of its operands is true.

The truth table for p OR q (also written as p ∨ q, Apq, p || q, or p + q) is as follows:

Logical disjunction

p q p ∨ q

T T T

T F T

F T T

F F F

Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q.

Logical implication

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

The truth table associated with the logical implication p implies q (symbolized as p ⇒ q, or more rarely Cpq) is as follows:

Logical implication

p q p ⇒ q

T T T

T F F

F T T

F F T

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

Material conditional

p q p → q

T T T

T F F

F T T

F F T

It may also be useful to note that p ⇒ q and p → q are equivalent to ¬p ∨ q.

Logical equality

Logical equality (also known as biconditional or exclusive nor) is an operation on two logical values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true.

The truth table for p XNOR q (also written as p ↔ q, Epq, p = q, or p ≡ q) is as follows:

Logical equality

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 same truth value (both true or both false), and false if they have different truth values.

Exclusive disjunction

Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true.

The truth table for p XOR q (also written as Jpq, or p ⊕ q) is as follows:

Exclusive disjunction

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

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

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

Logical NAND

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 a compound 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

The logical NOR is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of its operands are false. In other words, it produces a value of false if at least one of its operands is true. ↓ is also known as the Peirce arrow after its inventor, Charles Sanders Peirce, and is a Sole sufficient operator.

The truth table for p NOR q (also written as p ↓ q, or Xpq) is as follows:

Logical NOR

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 arguments p and q, 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 of De Morgan's laws