; Closure: For all a, b in A, the result of the operation a • b is also in A.

; Associativity: For all a, b and c in A, the equation ( a • b ) • c = a • ( b • c ) holds.

; Inverse element: For each a in A, there exists an element b in A such that a • b

; Commutativity: For all a, b in A, a • b = b • a.

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 example, the formula a AND b is satisfiable because one can find the values a = TRUE and b = TRUE, which make ( a AND b ) = TRUE.

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.

For example, in 1988 the United Church of Canada, that country's largest Protestant denomination, affirmed that " a ) All persons, regardless of their sexual orientation, who profess Jesus Christ and obedience to Him, are welcome to be or become full member of the Church ; and b ) All members of the Church are eligible to be considered for the Ordered Ministry.

For an edge type, b is perpendicular to the dislocation line, whereas in the cases of the screw type it is parallel.

For example, the Bourbaki group adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of ( a ) writing encyclopedic foundational works, and ( b ) supporting the axiomatic method as a research tool.

For example, suppose a and b are two sensible objects in a certain situation of affairs.

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

For the 2004 semi-final of the Eurovision Song Contest, staged in Istanbul thirty years after ABBA had won the contest in Brighton, Benny appeared briefly in a special comedy video made for the interval act, entitled " Our Last Video ".

For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period ( the interval between beats ) is half a second.

For a set of real variables lying within the interval M it has been shown that

For example in the consumer market, a typical motor oil change interval is around the.

For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word, specifically 1.

For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values ( such as an interval of non-zero length ) may be positive.

For example, if event A is choosing any number but 0. 5 from a uniform distribution on the unit interval, A is independent of itself, even though, tautologically, A fully determines A.

For example, the natural orientation of the interval gives an orientation of the two boundary points.

For a single channel, the signaling rate is given by, where SCSR is the single-channel signaling rate in bits per second, T is the minimum time interval in seconds for which each level must be maintained, and n is the number of significant conditions of modulation of the channel.

For a receiver, the attack-time delay is defined as the time interval from the instant a step radio-frequency ( RF ) signal, at a level equal to the receiver's threshold of sensitivity, is applied to the receiver input, to the instant when the receiver's output amplitude reaches 90 % of its steady-state value.

For a transmitter, the attack-time delay is defined as the interval from the instant the transmitter is keyed-on to the instant the transmitted RF signal amplitude has increased to a specified level, usually 90 % of its key-on steady-state value.

For an observation from the discrete component, the probability can either be written down directly or treated within the above context by saying that the probability of getting an observation in an interval that does contain a discrete component ( of being in interval j which contains discrete component k ) is approximately

For example, if X is the unit interval, not only is it possible to have a dense set of Lebesgue measure zero ( such as the set of rationals ), but it is also possible to have a nowhere dense set with positive measure.

For example, the set of all numbers satisfying is an interval which contains and, as well as all numbers between them.

For instance, when θ is restricted to π / 2 then sin ( θ ) is in this interval and arcsine is defined there.

For example, in the theory of quivers, the ( analogue of the ) unit interval is the graph whose vertex set is

For such a function, we can calculate its Fourier series on any interval that includes the points where ƒ is not identically zero.

For example, some marimbists feel the Musser-Stevens grip is more suitable for quick interval changes, while the Burton grip is more suitable for stronger playing or switching between chords and single-note melody lines.

For each of the 512 possible patterns, the rule table would state whether the center cell will be black or white on the next time interval.

For the Julia set is the line segment between-2 and 2, and the iteration corresponds to in the unit interval.

For example, the collection of all open intervals in the real line forms a base for a topology on the real line because the intersection of any two open intervals is itself an open interval or empty.

For Ahaz, the Scriptural data allow dating the beginning of his coregency with Jotham to some time in the six-month interval beginning of Nisan 1 of 735 BC.

For one thing, this is not a subject often discussed or analyzed.

For better or for worse, we all now live in welfare states, the organizing principle of which is collective responsibility for individual well-being.

For one thing, there is a natural belt of rock across the river bed ; ;

For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.

Even so astute a commentator as Harold Clurman of The Nation has said that `` Waiting For Godot '' is `` the concentrate of the contemporary European mood of despair ''.

For one thing, the world that Beckett sees is already shattered.

Harold Clurman is right to say that `` Waiting For Godot '' is a reflection ( he calls it a distorted reflection ) `` of the impasse and disarray of Europe's present politics, ethic, and common way of life ''.

For the beatnik, like the hipster, is in opposition to a society that is based on the repression of the sex instinct.

For this reason, too, their language is more forthright and earthy.

For the present it is enough to note that in the grotesque figure of Jacoby, at the moment of his collapse, all these elements come together in prophetic parody.

For Plato, `` imitation '' is twice removed from reality, being a poor copy of physical appearance, which in itself is a poor copy of ideal essence.

For both Plato and Aristotle artistic mimesis, in contrast to the power of dialectic, is relatively incapable of expressing the character of fundamental reality.

For Hammer, nothing is forbidden.

For example, suppose a man wearing a $200 watch, driving a 1959 Rolls Royce, stops to ask a man on the sidewalk, `` What time is it ''??

For the occasion on which everyone already knows everyone else and the host wishes them to meet one or a few honored newcomers, then the `` open house '' system is advantageous because the honored guests are fixed connective points and the drifting guests make and break connections at the door.

For this change is not a change from one positive position to another, but a change from order and truth to disorder and negation.

For paradigmatic history `` breaks '' rather than unfolds precisely when the movement is from order to disorder, and not from one order to a new order.

For this love of the boy for his mother is a hopeless and forbidden love, doomed by its nature.

For innocence, of all the graces of the spirit, is I believe the one most to be prayed for.

For what we propose, however, a psychoanalyst is not necessary, even though one aim is to enable the reader to get beneath his own defenses -- his defenses of himself to himself.

For this purpose a degree of intellectual and emotional involvement is necessary ; ;