Truth table
This page is the answer to the task Truth table in the Rosetta Code.
Contents
Description (from Rosetta Code)
A Truth table is a display of the inputs to, and the output of a Boolean equation organised as a table where each row gives one combination of input values and the corresponding value of the equation.
Task

Program
Explanation
Part 1. Input
λ is intended to be the boolean equation, given as a lambda expression.
Under Fōrmulæ, a lambda expression has two subexpressions, the first one is the set of variables to be mapped, and the second one is the mapping expression.
So λ_{1} is the left part of the lambda expression, this is, the set of variables. Under Fōrmulæ, counting starts in one, rather than zero as many programming languages.
The expression x is the cardinality of x, so λ_{1} is the number of variables.
Part 2. Cartesian product exponentiation
The expression
Is the Cartesian product exponentiation of the list {true, false}. See the following examples.
Please note that the cartesian product exponentiation always produces a list of lists, shown here with a matrix notation.
This step produces the combination of logic values (true and false) according to the number of variables of the boolean equation.
Part 3. Arraization
The Fōrmulæ arraization expression is similar in notation and functionality to summation, see the following examples:
But in here, we use a variation that iterates over the elements of a list:
Part 4. Calculation and output
The expression
Creates a array from the elements of the values array. Note that this array is a list of lists (a matrix), produced as a cartesian product exponentiation, so the v symbol repeatedly acquires an array (for every row of the matrix).
The expression
Is the lambda application of the λ lambda expression using the set of values provided precisely as the symbol v.
Note that the arraization produces an array, and the expression to create the elements of the array is also an array, the final result is shown as a matrix.
Hacking mode
The provided program was created with clarity purposes in mind. A compressed, but not so clear version could be:
Case 1. One variable
The following example produces the logical negation table
Case 2. Two variables
The following example produces the logical conjunction table
Because there is no restrictions about the mapping expression, it can be an array of expressions involving the arguments.
The following example produces the truth table for logical conjunction, disjunction, material implication, equivalence, and exclusive disjunction.
Case 3. Three variables
In the following example, the truth table is used to show that a boolean formula is a tautology.
Because any function is a named lambda expression, the example also shows the passing of a function as an argument to another function.
See also
Finitevalued logic example.