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If S contains an upper bound then that upper bound is unique and is called the greatest element of S. The greatest element of S ( if it exists ) is also the least upper bound of S. A special situation does occur when a subset is equal to the set of lower bounds of its own set of upper bounds.
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If and S
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )
* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =
* If a set S is in M and S is congruent to T then T is also in M and a ( S )
* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )
* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.
If and contains
If the tree contains data, we can simply simultaneously store it in a consecutive array in preorder.
If the cord is twisted up tightly on itself and then the bucket is released, it begins to spin rapidly, not only with respect to the experimenter, but also in relation to the water it contains.
In the words of Henry Chadwick, " If the Consolation contains nothing distinctively Christian, it is also relevant that it contains nothing specifically pagan either ... is a work written by a Platonist who is also a Christian, but is not a Christian work.
If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.
If each pixel on the display actually contains three rectangular subpixels of red, green, and blue, in that fixed order, then things on the screen that are smaller than one full pixel in size can be rendered by lighting only one or two of the subpixels.
If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.
If the frame contains areas where nothing has moved, the system simply issues a short command that copies that part of the previous frame, bit-for-bit, into the next one.
If both copies are damaged, the Purkinje layer ( a part of the cerebellum that contains better-connected neurons than any other ) develops abnormally, runting is more common, and pups die within weeks due to inadequate lung development.
If neither A nor B includes the idea of existence, then " some A are B " simply adjoins A to B. Conversely, if A or B do include the idea of existence in the way that " triangle " contains the idea " three angles equal to two right angles ", then " A exists " is automatically true, and we have an ontological proof of A's existence.
If a project plan contains a significant portion of LOE, and the LOE is intermixed with discrete effort, EVM results will be contaminated.
If an explosive molecule contains just enough oxygen to convert all of its carbon to carbon dioxide, all of its hydrogen to water, and all of its metal to metal oxide with no excess, the molecule is said to have a zero oxygen balance.
If a map contains a reducible configuration, then the map can be reduced to a smaller map.
If the packet contains options, this field contains the option type of the next option.
# If V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.
The inscription on his tomb contains Persian words penned which state: اگرفردوس روی زمین است همین است و همین است و همین است ( If there is a paradise on earth, it is this, it is this, it is this!
If the picture contains an area of the same color, it can be compressed without loss by saying " 200 red dots " instead of " red dot, red dot, ...( 197 more times )..., red dot.
If each group contains less than half the population, then some of the population is exactly equal to the median.
If S contains two elements that are not pairwise orthogonal ( in particular, the set of all quantum states includes such pairs ) then an argument like that given above shows that the answer is no.
If each additional layer contains less oil, the final painting will crack and peel.
# If he contains and sustains all things, would he not be existence itself?
If it is made from durum wheat, the pasta contains protein and gluten.
If and upper
If it wants to become a high-income or upper middle-income country, it can not do so with this kind of economic structure.
If the destination background is opaque, then, and if you enter it to the upper equation:
If the symbol width is n, the codes of width n + 1 fall naturally into two blocks: the lower block of 2 < sup > n </ sup > codes for coding single symbols, and the upper block of 2 < sup > n </ sup > codes that will be used by the decoder for sequences of length greater than one.
If the player wishes to play the upper 8 ' from the upper manual only and not from the lower manual, a stop handle disengages the jacks labeled A and engages instead an alternative row of jacks called " lute stop " ( not shown in the Figure ).
If a trill is made with the tip of the tongue against the upper gum, it is called an apical ( tongue-tip ) alveolar trill ; the IPA symbol for this sound is.
If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.
* If the upper cervical cord is involved, all four limbs may be involved and there is risk of respiratory paralysis ( segments C3, 4, 5 to diaphragm ).
That means, define E ≤ F if E is an initial segment of F and the ordering of the members in E is the same as their ordering in F. If E is a chain in A, then the union of the sets in E can be ordered in a way that makes it a continuation of any set in E ; this ordering is a well ordering, and therefore, an upper bound of E in A.
* If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components, B < f < sub > s </ sub >/ 2, of the signal to allow for perfect reconstruction.
If the upper air is warmer than predicted by the adiabatic lapse rate (), then when a parcel of air rises and expands, it will arrive at the new height at a lower temperature than its surroundings.
If, on the contrary, the upper air is cooler than predicted by the adiabatic lapse rate, then when the air parcel rises to its new height it will have a higher temperature and a lower density than its surroundings, and will continue to accelerate upward.
If the C major chord was orchestrated for the trumpets and trombones playing fortissimo in their upper registers, it would sound very bright ; But if the same chord was orchestrated for the celli and string basses playing sul tasto, doubled by the bassoons and bass clarinet, it might sound heavy and dark.
If the hero gets the upper hand, he becomes the Sun, his wife the Moon.
If the upper triangle is used without the lower one, it designates a " half-long " vowel.
If a muscle has impaired function following an upper motor neuron lesion, other changes such as increased muscle stiffness are likely to affect the feeling of resistance to passive stretch.
If Mario can reach the upper access, there is a bonus level in which the player can win extra lives or power-ups.
If pupils follow the average pattern of school attendance, pupils will be in " year 12 " between the ages of 16 and 17, and " year 13 " between 17 and 18, however many schools still refer to them as the " lower sixth " and " upper sixth " or " AS " and " A2 ".
If only the area above the upper lip is left unshaven, the resulting facial hairstyle is known as a moustache ; if hair is left only on the chin, the style is a chin beard.
If the reactor coolant pumps are not running, the upper part of the core will be exposed to a steam environment and heatup of the core will begin.
If the belt has existed for longer than this period, which appears likely, it imposes an upper limit on Epsilon Eridani b's eccentricity of about 0. 10 – 0. 15.
0.337 seconds.