Everything about Truth Table totally explained
A
truth table is a
mathematical table used in
logic — specifically in connection with
Boolean algebra,
boolean functions, and
propositional calculus — to compute the functional values of logical
expressions on each of their functional arguments, that is, on each combination of values taken by their logical variables. In particular,
truth tables can be used to tell whether a propositional expression is true for all legitimate input values, that is,
logically valid.
"The pattern of reasoning that the truth table tabulates was
Frege's,
Peirce's, and
Schröder's by 1880. The tables have been prominent in literature since 1920 (
Lukasiewicz,
Post,
Wittgenstein)" (
Quine, 39).
Lewis Carroll had formulated truth tables as early as 1894 to solve certain problems, but his manuscripts containing his work on the subject were not discovered until 1977
(External Link
). Wittgenstein's
Tractatus Logico-Philosophicus uses them to place
truth functions in a series. The wide influence of this work led to the spread of the use of truth tables.
Truth tables are used to compute the values of propositional expressions in an effective manner that's sometimes referred to as a
decision procedure. A propositional expression is either an
atomic formula — a propositional constant,
propositional variable, or propositional function term (for example,
Px or
P(
x)) — or built up from atomic formulas by means of logical operators, for example,
AND (
),
OR (
),
NOT (
). For instance,
is a propositional expression.
The column headings on a truth table show (i) the propositional functions and/or variables, and (ii) the truth-functional expression built up from those propositional functions or variables and operators. The rows show each possible valuation of
T or
F assignments to (i) and (ii). In other words, each row is a distinct interpretation of (i) and (ii).
Truth tables for
classical logic are limited to
Boolean logical systems in which only two
logical values are possible,
false and
true, usually written
F and
T, or sometimes
0 or
1, respectively.
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 or
¬p) is as follows:
Logical Negation>
| p |
¬p |
| F |
T |
| T |
F |
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 for
p AND q (also written as
p ∧ q,
p & q, or
pq) 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
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 for
p OR q (also written as
p ∨ q,
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 |
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, that produces a value of
false just in the singular case the first operand is true and the second operand is false.
The truth table associated with the material conditional
not p or 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 |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
F |
T |
Logical equality
Logical equality (also known as biconditional) 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 for
p EQ q (also written as
p = q,
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 |
Exclusive disjunction
Exclusive disjunction is an
operation on two
logical values, typically the values of two
propositions, that produces a value of
true if and only if one but not both of its operands is true.
The truth table for
p XOR q (also written as
p + q,
p ⊕ q, or
p ≠ q) is as follows:
Exclusive Disjunction>
| p |
q |
p ⊕ q |
| T |
T |
F |
| T |
F |
T |
| F |
T |
T |
| F |
F |
F |
XOR can also be written as (p = 1 ∧ q = 0)∨ (p = 0 ∧ q = 1).
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 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 for
p NAND q (also written as
p | q or
p ↑ q) is as follows:
Logical NAND>
| p |
q |
p ↑ q |
| T |
T |
F |
| T |
F |
T |
| F |
T |
T |
| F |
F |
T |
In the case of logical NAND, it's clearly expressible as a compound of NOT and AND.
The negation of conjunction
:
XNOR (exclusive nor).
Johnston diagrams, similar to
Venn diagrams and
Euler diagrams, provide a way of visualizing truth tables. An interactive
Johnston diagram illustrating truth tables is at
LogicTutorial.com
Condensed truth tables for binary operators
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:
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
Truth tables are also used to specify the functionality of
hardware look-up tables (LUTs) in
digital 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 a
bit in a
binary number, truth table values can be efficiently encoded as
integer values in
electronic 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 index
k based on the input values of the LUT, in which case the LUT's output value is the
kth bit of the integer. For example, to evaluate the output value of a LUT given an
array of
n boolean input values, the bit index of the truth table's output value can be computed as follows: if the
ith input is true, let V
i = 1, else let V
i = 0. Then the
kth bit of the binary representation of the truth table is the LUT's output value, where
k = V0*2^0 + V1*2^1 + V2*2^2 + ... + V
n*2^
n.
Truth tables are a simple and straightforward way to encode boolean functions, however given the
exponential growth in size as the number of inputs increase, they're not suitable for functions with a large number of inputs. Other representations which are more memory efficient are text equations and
binary decision diagrams.
Applications of truth tables in digital electronics
In digital electronics (and computer science, fields of engineering derived from applied logic and math), truth tables can be used to reduce basic boolean operations to simple correlations of inputs to outputs, without the use of logic gates or code. For example, a binary addition can be represented with the truth table:
A B | C R
1 1 | 1 0
1 0 | 0 1
0 1 | 0 1
0 0 | 0 0
where
A = First Operand
B = Second Operand
C = Carry
R = 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.
Note that this table doesn't describe the logic operations necessary to implement this operation, rather it simply specifies the function of inputs to output values.
In this case it can only be used for very simple inputs and outputs, such as 1's and 0's, 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 2x2, or four. So the result is four possible outputs of C and R. If one was to use base 3, the size would increase to 3x3, 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 a
full adder's logic:
A B C* | C R
0 0 0 | 0 0
0 1 0 | 0 1
1 0 0 | 0 1
1 1 0 | 1 0
0 0 1 | 0 1
0 1 1 | 1 0
1 0 1 | 1 0
1 1 1 | 1 1
Same as previous, but..
C* = Carry from previous adder
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