Tutorials 242 1 Which of the followings define a binary operations on the set

Tutorials 242 1 which of the followings define a

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Tutorials 2.4.2. (1) Which of the followings define a binary operations on the set of integers? Of those that do, which are associative? Which are commuta- tive? Which are idempotent? (a) m n = mn + 1 , (b) m n = ( m + n ) ÷ 2 , (c) m n = m, (d) m n = mn 2 , (e) m n = m 2 + n 2 , (f) m n = 3 , (g) m n = m n . (2) Let m be a binary operation on the set of real numbers defined by x y = ax + by + c. (a) Find the set of all triples ( a, b, c ) of real numbers, for which is associative. (b) Find the set of all triples ( a, b, c ) of real numbers, for which is commuta- tive. 56
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(c) Find the set of all triples ( a, b, c ) of real numbers, for which is idempotent. (3) Find all binary operations on the set { 0 , 1 } . 2.4.2 Algebraic structures Remarks 2.4.3. Intuitively, a mathematical structure is a pair ( X, s ) , where X is a set, called the underlying set of the structure ( X, s ) , and s; the struc- ture itself, is anything defined on X. Sometimes we shall write just X instead of ( X, s ) , or, say, if s itself is a pair ( s 1 , s 2 ) , then consider ( X, s ) as the triple ( X, s 1 , s 2 ) , etc. An isomorphism f : ( X, s ) ( X 0 , s 0 ) of structures of the same type is a bi- jection f : X X 0 such that both f and its inverse bijection preserve the struc- ture. When such an isomorphism exist, we also say that ( X, s ) and ( X 0 , s 0 ) are isomorphic to each other. We shall not give any general definition, but only consider a few examples: 2.4.3 Examples of algebraic structures 1. A magma is a pair ( X, ) , where X is a set and is a binary operation on it. An isomorphism f : ( X, ) ( X 0 , 0 ) of magmas is a bijection f : XrightarrowX 0 with f ( x y ) = f ( x ) 0 f ( y ) * forall x and y in X : Note that * implies that the bijection g ; inverse of f ; has the same property. Indeed, for every x 0 and y 0 in X 0 , we have g ( x 0 0 y 0 ) = g ( fg ( x 0 ) 0 fg ( y 0 )) = gf ( g ( x 0 ) 0 g ( y 0 )) = g ( x 0 ) 0 g ( y 0 ) . Sometimes one uses either the multiplicative notation, writing x · y (or simply xy ) instead of x y , or the additive notation, writing x + y , and then calls magmas multiplicative or additive, respectively. If so, * becomes f ( xy ) = f ( x ) f ( y ) or f ( x + y ) = f ( x ) + f ( y ) , accordingly. In old literature magmas were called groupoids. 57
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2. A magma ( X, ) is said to be associative, commutative, idempotent, if so is the operation , respectively. An associative magma is also called a semigroup, and, accordingly, an associative commutative magma is called a commutative semigroup, and term ”idempotent” is used in the same way. One also says that an element x of a magma ( X, ) is idempotent if x x = x, according to this terminology, a magma ( X, ) is idempotent if and only if so is every element of X. 3. A unitary magma is a triple ( X, e, m ), in which ( X, m ) is a magma and e an element in X with m ( e, x ) = x = m ( x, e ) for every x in X : When either the multiplicative, or the additive notation is used, one writes 1 or 0 instead of e and 1 x = x = x 1 or 0 + x = x = x + 0 , respectively. The element e above sometimes called the neutral element’ or the identity element or the unit, in the case of multiplicative notation, or the zero element or just zero, in the case of additive notation.
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