For example, if the domain is the set of all real numbers, one can assert in first-order logic the existence of an additive inverse of each real number by writing ∀ x ∃ y ( x + y = 0 ) but one needs second-order logic to assert the least-upper-bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum.

# there is an x such that x is presently King of France: ∃ x ( using ' PKoF ' for ' presently King of France ')

These are dual because ∃ x .¬ P ( x ) and ¬∀ x. P ( x ) are equivalent for all predicates P: if there exists an x for which P fails to hold, then it is false that P holds for all x.

* ∃ x ∀ y ∃ z ( H ( x, y, x ))

* ∀ x ∃ z ( F ( x ) ∧ G ( a, z ))

* ∃ x ∀ y ∃ z ( G ( a, z ) ∨ H ( x, y, z ))</ big >

* < big >∃ x ∀ y ∃ z ( H ( x, y, z ))</ big >: " Somebody made everybody hit somebody.

* < big >∃ x ∀ y ∃ z ( G ( a, z ) ∨ H ( x, y, z ))</ big >: " Either Socrates hates somebody or somebody made everybody hit somebody.

* ∃ x ∃ y ( M ( a, b, x ) ∧ M ( a, c, y )): Father Ted officiated at the marriage of Jack to somebody and Father Ted officiated at the marriage of Jill to somebody.

∃ x + ∃ y

* ∃ x ∃ y = ∃( x ∃ y ).

For example, the formula ∀ a ∀ b ∃ c / b ∃ d / a φ ( a, b, c, d ) (" x / y " should be read as " x independent of y ") cannot be expressed in FOL.

For a language L containing ¬ (" not "), ∧ (" and "), ∨ (" or ") and quantifiers (∀ " for all " and ∃ " there exists "), Tarski's inductive definition of truth looks like this:

* < big >∀ x ∃ z ( F ( x ) ∧ G ( a, z ))</ big >: " Everybody is sleeping and Socrates hates somebody.

Two common quantifiers are the existential ∃ (" there exists ") and universal ∀ (" for all ") quantifiers.

It is denoted by the logical operator symbol ∃ ( pronounced " there exists " or " for some "), which is called the existential quantifier.

if ∃ v s. t.

else if ∃ e s. t.

else if ∃ v s. t.

else if ∃ v s. t.

The basic duality of this type is the duality of the ∃ and ∀ quantifiers.

∃ x is the existential closure of x. Dual to ∃ is the unary operator ∀, the universal quantifier, defined as ∀ x := (∃ x ' )'.

A monadic Boolean algebra has a dual formulation that takes ∀ as primitive and ∃ as defined, so that ∃ x := (∀ x ' )'.

In a 3-way ANOVA with factors x, y and z, the ANOVA

model includes terms for the main effects ( x, y, z ) and

Some adaptations of the Latin alphabet are augmented with ligatures, such as æ in Old English and Icelandic and Ȣ in Algonquian ; by borrowings from other alphabets, such as the thorn þ in Old English and Icelandic, which came from the Futhark runes ; and by modifying existing letters, such as the eth ð of Old English and Icelandic, which is a modified d. Other alphabets only use a subset of the Latin alphabet, such as Hawaiian, and Italian, which uses the letters j, k, x, y and w only in foreign words.

The aspect ratio is expressed as two numbers separated by a colon ( x: y ).

A common misunderstanding is that x and y represent actual length and height.

: When comparing the above illustration to the below text, please note that the above x: y aspect ratio values are shown as vertical orientation rectangles to better demonstrate visual differences, whereas the aspect ratio values of the text below are written as rotated horizontal orientation rectangles ( e. g. compare 3: 4 vertical orientation illustration to 4: 3 horizontal orientation text ).

Aspect ratios are mathematically expressed as x: y ( pronounced " x-to-y ") and x × y ( pronounced " x-by-y "), with the latter particularly used for pixel dimensions, such as 640 × 480.

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

When the boot loader detected a CP / M floppy, the Aster would reconfigure its internal memory architecture on the fly to optimally support CP / M with 60 KB free RAM for programs ( TPA ) and an 80 x 25 display.

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

for all x and y in M.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

Changing x to x / b stretches the graph horizontally by a factor of b. ( think of the x as being dilated )

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

The coefficient a in the term of x < sup > b </ sup > y < sup > c </ sup > is known as the binomial coefficient or ( the two have the same value ).

is a multiple of d. x and y are called Bézout coefficients for ( a, b ); they are not unique.

For given nonzero integers a and b there is a nonzero integer of minimal absolute value among all those of the form ax + by with x and y integers ; one can assume d > 0 by changing the signs of both s and t if necessary.

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

Ax = b gives a bound on how inaccurate the solution x will be after approximation.

In particular, one should think of the condition number as being ( very roughly ) the rate at which the solution, x, will change with respect to a change in b. Thus, if the condition number is large, even a small error in b may cause a large error in x.

On the other hand, if the condition number is small then the error in x will not be much bigger than the error in b.

The condition number is defined more precisely to be the maximum ratio of the relative error in x divided by the relative error in b.

The equation of a circle is ( x − a )< sup > 2 </ sup > + ( y − b )< sup > 2 </ sup > = r < sup > 2 </ sup > where a and b are the coordinates of the center ( a, b ) and r is the radius.