If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

If f is a surjection and a ~ b ↔ f ( a ) = f ( b ), then g is a bijection.

If ~ and ≈ are two equivalence relations on the same set S, and a ~ b implies a ≈ b for all a, b ∈ S, then ≈ is said to be a coarser relation than ~, and ~ is a finer relation than ≈.

If negation is cyclic and "∨" is a " max operator ", then the law can be expressed in the object language by ( P ∨ ~ P ∨ ~~ P ∨ ... ∨ ~...~ P ), where "~...~" represents n − 1 negation signs and "∨ ... ∨" n − 1 disjunction signs.

If G is a group ( with identity element e and operation *) and ~ is a binary relation on G, then ~ is a congruence whenever:

If it is assumed that the calculation occurs near room temperature (~ 300 K ) the Von Neumann-Landauer Limit can be applied to estimate the energy required as ~ 10 < sup > 18 </ sup > joules, which is equivalent to consuming 30 gigawatts of power for one year.

The text " If ye break faith ~ we shall not sleep " and " Buy Victory Bonds " are written at the top and bottom respectively.

If R is an integral domain, define a relation ~ on nonzero fractional ideals of R by I ~ J whenever there exist nonzero elements a and b of R such that ( a ) I = ( b ) J.

If a typical mammalian cell ( diameter ~ 10 micrometre ) were magnified to the size of a watermelon (~ 1 ft / 30 cm ), the lipid bilayer making up the plasma membrane would be about as thick as a piece of office paper.

If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X /~ has the diffeology generated by all compositions of plots of X with the projection from X to X /~.

* Operation High Time ( 1969 )-( Sime ~ Gen Universe )-Worlds of If Magazine, January

If f is a rational function of degree d, then T ( r, f ) ~ d log r ; in fact, T ( r, f ) = O ( log r ) if and only if f is a rational function.

If the operator < tt >~</ tt > has left associativity, this expression would be interpreted as < tt >( a ~ b ) ~ c </ tt > and evaluated left-to-right.

If the operator has right associativity, the expression would be interpreted as < tt > a ~ ( b ~ c )</ tt > and evaluated right-to-left.

* Kaede ~ If Trans ...~, a VHS released by Japanese rock band Dir en grey

If the conductivity for oxygen ions in SOFC can remain high even at lower temperature ( current target in research ~ 500 ° C ), material choice for SOFC will broaden and many existing problems can potentially be solved.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If there were a ' zero ' symbol, unary would effectively be a binary system.

If any official operation abroad begins to go wrong, we shall look to the ambassador to find out why and to get suggestions for remedial action.

If that's done, the house can be designed and oriented for best operation, and this can mean savings both in the size of equipment and in the cost of the house itself.

If the resulting four kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit ( i. e. 1 kilobyte ) table by the use of circular rotates.

* 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:

If f is not a function, but is instead a partial function, it is called a partial operation.

* If the operation is associative, ( ab ) c = a ( bc ), then the value depends only on the tuple ( a, b, c ).

* If the operation is commutative, ab = ba, then the value depends only on

If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.

If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

" If you look at it as a whole ," a senior S / LPD official said, " the Office of Public Diplomacy was carrying out a huge psychological operation of the kind the military conducts to influence a population in denied or enemy territory.

If these relations are stored physically as separate disk files, completing a database query that draws information from several relations ( a join operation ) can be slow.

If this operation has to be done in real time video games there is an easy trick to boost performance.

# 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.

If the reader evaluates an expression but does not follow the correct order of operation, the reader will come forth with a different value.

If predictable operation is desired, negative feedback is used, by applying a portion of the output voltage to the inverting input.

If one could achieve reliable operation of VLB at 50mhz it was extremely fast-but again this was notoriously difficult to achieve, and often it was discovered not to be possible with a given hardware configuration.

If the weather is below VMC, pilots are required to use instrument flight rules, and operation of the aircraft will primarily be through referencing the instruments rather than visual reference.

If the check in operation succeeds, then the version numbers of all files involved automatically increment, and the CVS-server writes a user-supplied description line, the date and the author's name to its log files.

If an operation is to be performed on the whole structure, each node calls the operation on its children ( recursively ).

If the annular muscles are contracted, the body cross-section is reduced, and the corresponding segment stretches, since its volume must remain constant due to the incompressible behaviour of the coelom's liquid contents ; this is the usual mode of operation of the hydrostatic skeleton as also employed by other worms.

If ( G ,*) is a group, we define its opposite group ( G < sup > op </ sup >,*< sup > op </ sup >) as follows: G < sup > op </ sup > is the same set as G, and the operation *< sup > op </ sup > is defined by.

If the operation is associative then if an element has both a left inverse and a right inverse, they are equal.