Associativity
In mathematics, a binary operation * on a set S is called associative if for all x, y and z in S, (x * y) * z = x * (y * z).The most commonly known examples of associativity are addition and multiplication of natural numbers; for example:
- (7 + 3) + 9 = 7 + (3 + 9), since the expression on the left evaluates to 10 + 9 = 19, which the expression on the right evaluates to 7 + 12 = 19, the same value;
- (10 × 5) × 3 = 10 × (5 × 3), since the expression on the left evaluates to 50 × 3 = 150, while the expression on the right evalutes to 10 × 15 = 150.
Other examples of associative binary operations include addition and multiplication of real numbers, complex numbers and square matrices; addition of vectors; and intersection and union of sets. Also, 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.
A set with an associative binary operation on it is called a semigroup; monoids and groups are examples of semigroups.
See also Commutativity, Distributive property, Identity element
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absolutely! There's no music in his smile. Oh! you should see a
it not. None of you English have. You go so.'
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myself--Where am I? Among what boors have I fallen? But Evan is no
his shoulders properly, and to direct his eyes--Oh! the eyes! you
soul! Can any of you look the unutterable without being absurd! You
a sheep might yawn.
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qualities to forgive him, poor man! And that stiff bandbox of a man of
were all angles and sections, and were taken to pieces every night and
will--but, Diacho! to be married to that! He is not civilized. None of
many intrusive oxen--absolutely! One of your men trod on my toe the
the half of him forward--I thought he was going to break in two--then
whether he didn't say, MARM!'
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