** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

* The Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

For example, in most systems of logic ( but not in intuitionistic logic ) Peirce's law ((( P → Q )→ P )→ P ) is a theorem.

: For the theorem of propositional logic which expresses Disjunction elimination, see Case analysis.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

For instance, if we impose some requirements on a distribution f, we can attempt to translate these requirements in terms of the Fourier transform of f. The Paley – Wiener theorem is an example of this.

For compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.

For a discussion of the name " Mittag-Leffler " in its relation with the Mittag-Leffler theorem, see this thread on MathOverflow.

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

For a closely related theorem see the Bing metrization theorem.

For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem ( where x, y, z ... are the free variables in the proposition p ).

For, in 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals ( now referred to as the Abel – Ruffini theorem ).

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

For Avicenna ( Ibn Sina ), for example, the a tabula rasa is a pure potentiality that is actualized through education, and knowledge is attained through " empirical familiarity with objects in this world from which one abstracts universal concepts " developed through a " syllogistic method of reasoning in which observations lead to propositional statements which when compounded lead to further abstract concepts.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.

For example, both philosophers of language and semanticists make use of propositional, predicate and modal logics to express their ideas about word meaning ; what Frege termed ' sense '.

For example, if φ and ψ are propositional constants and → is a binary operator, then φ →( φ → ψ ) is a proposition, which might also be written as → φ → φψ or in another order.

For example, if P ( x ) is the propositional function " x is between 0 and 1 ", then, for a domain of discourse X of all natural numbers, the existential quantification " There exists a natural number x which is between 0 and 1 " is symbolically stated:

For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises does not depend on these data.

For example, the propositional Horn clause written above behaves as the procedure:

: For the rule of inference of propositional logic based on the same concept, see double negative elimination.

For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid.

For most types of propositional

( For more on von Wright's departure from and return to the syntax of the propositional calculus, see Deontic Logic: A Personal View and A New System of Deontic Logic, both by Georg Henrik von Wright.

For example ( using a propositional attitude clause ), if one quantifies into the statement " Ralph believes that Ortcutt is a spy ," the result is ( partly formalized ):

For example, in a given propositional logic, we might define a formula as follows:

For the purposes of the propositional calculus, propositions ( utterances, sentences, assertions ) are considered to be either simple or compound.

For the purposes of the propositional calculus a compound proposition can usually be reworded into a series of simple sentences, although the result will probably sound stilted.

But suppose we put it thus ' For all p, if he asserts p, p is true ', then we see that the propositional function p is true is simply the same as p, as e. g. its value ' Caesar was murdered is true ' is the same as ' Caesar was murdered '.

For propositional connectives, this is easy ; one simply applies the corresponding Boolean operators to the truth values of the subformulae.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.

For Alexander Gottlieb Baumgarten aesthetics is the science of the sense experiences, a younger sister of logic, and beauty is thus the most perfect kind of knowledge that sense experience can have.

For instance, a professor of formal logic called Chin Yueh-lin – who was then regarded as China ’ s leading authority on his subject – was induced to write: “ The new philosophy Marxism-Leninism, being scientific, is the supreme truth ”.

For classical logic, it can be easily verified with a truth table.

For Husserl this is not the case: mathematics ( with the exception of geometry ) is the ontological correlate of logic, and while both fields are related, neither one is strictly reducible to the other.

For example, he coined the programming phrase " two or more, use a for ," alluding to the rule of thumb that when you find yourself processing more than one instance of a data structure, it is time to consider encapsulating that logic inside a loop.

For a history of first-order logic and how it came to be the dominant formal logic, see José Ferreirós 2001.

For the life of me I cannot understand the logic of having a Governor who is part-time and doesn ’ t live at Government House.

For a concise description of the symbols used in this notation, see List of logic symbols.

For a concise description of the symbols used in this notation, see List of logic symbols.

For higher speed, the resistors used in RTL were replaced by diodes, leading to diode-transistor logic ( DTL ).

For small-scale logic, designers now use prefabricated logic gates from families of devices such as the TTL 7400 series by Texas Instruments and the CMOS 4000 series by RCA, and their more recent descendants.

For thousands of years, society considers Mentats the embodiment of logic and reason.

For example, the AT91CAP from Atmel has a block of logic that can be customized during manufacturer according to user requirements.

For data transmission lines ( TxD, RxD and their secondary channel equivalents ) logic one is defined as a negative voltage, the signal condition is called marking.

For any given level of general performance, a RISC chip will typically have far fewer transistors dedicated to the core logic which originally allowed designers to increase the size of the register set and increase internal parallelism.

For the skeptics, such logic was thus an inadequate measure of truth and could create as many problems as it claimed to have solved.

: For example, the design of a multitier architecture is made simple using Java Servlets in the web server and various CORBA servers containing the business logic and wrapping the database accesses.