Search results
Results from the WOW.Com Content Network
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. [1]
A monopolist's total revenue can be graphed as in Figure 1, in which Price (P) is the height of the box, and Quantity (Q) is the width. P × Q {\displaystyle P\times Q} , (i.e., total revenue) equals the area of the box.
Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; [75] and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). [51]
The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p ∨ s) = q, then let p = q. Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus impredicative. Either of two conditions can result: [25] oscillation or memory.
The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However, it cannot be formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory.
However, when a value is assigned to x, such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false. Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):
The Church numeral n is a function that takes a function f as argument and returns the n-th composition of f, i.e. the function f composed with itself n times. This is denoted f ( n ) and is in fact the n -th power of f (considered as an operator); f (0) is defined to be the identity function.
This truth definition requires that one must select a variable assignment function (μ above) before truth values for even atomic formulas can be defined. Then the truth value of a sentence is defined to be its truth value under any variable assignment, and it is proved that this truth value does not depend on which assignment is chosen.