** 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 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.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

In 1997 Enya released her greatest hits collection, Paint the Sky with Stars: The Best of Enya, again a top five smash in the UK and Germany, which featured two new songs: " Paint the Sky with Stars " and " Only If ..."; " Only If ..." later became a single.

If we reduce the larger number by subtracting the smaller one from it, the greatest common divisor does not change.

If two numbers have no prime factors in common, their greatest common divisor is 1 ( obtained here as an instance of the empty product ), in other words they are coprime.

If a and b are not both zero, the greatest common divisor of a and b can be computed by using least common multiple ( lcm ) of a and b:

If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

The song is included on Dixon's 1992 greatest hits album ( If ) I'm A Ham, Well You're A Sausage.

If false, it may reflect a desire to silence one of the greatest embarrassments to the Bush administration.

In a June 1918 leaflet entitled " Appeal for Enlistment ", Gandhi wrote " To bring about such a state of things we should have the ability to defend ourselves, that is, the ability to bear arms and to use them ... If we want to learn the use of arms with the greatest possible despatch, it is our duty to enlist ourselves in the army.

According to Mel Hein, " If he hadn't been killed, he could have been the greatest tackle who ever played football.

"… they are obliged to live in a separate part of town …; for they are considered as unclean creatures … Under the pretext of their being unclean, they are treated with the greatest severity and should they enter a street, inhabited by Mussulmans, they are pelted by the boys and mobs with stones and dirt … For the same reason, they are prohibited to go out when it rains ; for it is said the rain would wash dirt off them, which would sully the feet of the Mussulmans … If a Jew is recognized as such in the streets, he is subjected to the greatest insults.

If no individual had a majority, then the House of Representatives would choose from the five individuals with the greatest number of electoral votes.

If multiple individuals are tied for second place, the Senate may consider all of them, in addition to the individual with the greatest number of votes.

If so, all I can say is, it is a new Liberalism, and not the one that I have known and practised under more illustrious auspices than these, under one who was not merely the greatest Liberal but the greatest financier that this country has ever known — I mean Mr. Gladstone ... Gladstone ranks as the great financial authority of our country ... Mr. Gladstone would be 100 in December if he were alive, but, centenarian as he would be, I am inclined to think that he would make very short work of the deputation of the Cabinet that waited on him with this measure, and that they would soon find themselves on the stairs, if not in the street.

In analysis the infimum or greatest lower bound of a subset S of real numbers is denoted by inf ( S ) and is defined to be the biggest real number that is smaller than or equal to every number in S. If no such number exists ( because S is not bounded below ), then we define inf ( S ) = −∞.

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.