BINARY OPERATORS-(E)-03-Theory

9.2.1 Definition. A binary operation on a non-empty set A is a mapping which associates with each ordered pair (a, b) of elements of A, a uniquely defined element c  A. This is a mapping from the product set A × A to A. Symbolically, a map : A  A  A, is called a binary operation on the set A. The image of the element (a, b)  A  A is denoted by a * b. If a set A is closed with respect to the composition, then we say that * is a binary operation on the set A. Let, a  N, b  N  a + b  N for all a, b  N. Multiplication on N is also a binary operation, since a  N, b  N  a  b  N for all a, b  N But subtraction on N is not a binary operation, since 3  N, 5  N but 3 – 5 = – 2  N. Note :  It is obvious that addition as well as multiplication are binary operations on each one of the sets Z (of integer), Q (of rational number), R (of real number) and C (of all complex number).  Subtraction is a binary operation on each of the sets Z, Q, R and C. But it is not binary operation on N.  Division is not a binary operation on any of sets N, Z, Q, R and C. 9.2.2 Types of Binary Operation (1) Commutative binary operation : A binary operation * on a set S is said to be commutative if a * b = b * a for all a, b  S Addition and multiplication are commutative binary operations on Z but the subtraction is not a commutative binary operation, since 2 – 3  3 – 2. (2) Associative binary operation : A binary operation * on a set S is said to be associative if (a * b) * c = a * (b * c) for all a, b, c  S Addition and multiplication are associative binary operations on N, Z, Q, R and C. But subtraction is not an associative binary operation on Z, Q, R and C. (3) Distributive binary operation : Let * and o be two binary operations on a set S. Then * is said to be (i) Left distributive over o if a * (b o c) = (a * b) o (a * c) for all a, b, c  S; (ii) Right distributive over o if (b o c) * a = (b * a) o (c * a) for all a, b, c  S. If * is both left and right distributive over o, then * is said to be distributive over o. Example : The multiplication () on Z is distributive over addition (+) on Z, since a  (b + c) = a  b + a  c and (b + c)  a = b  a + c  a for all a, b, c  Z. But addition is not distributive over multiplication. 9.2.3 Identity and Inverse elements (1) Identity element : Let * be a binary operation on a set S. An element e  S is said be an identity element for the binary operation * if a * e = a = e * a for all a  S. For addition on Z, 0 is the identity element, since a + 0 = a = 0 + a for all a  Z. For multiplication on R, 1 is the identity element, since 1  a = a = a × 1 for all a  R. (2) Inversible element for a binary operation with identity : An element a of a set A is said to be inversible for a binary operation * with identity e if  b  A such that a * b = e = b * a. Also, then b is said to be an inverse of a and is denoted by a–1. The inversible elements in A are also called the units in A. The identity element is always inversible and is its own inverse, since e * e = e * e = e. Thus e–1 = e. 9.2.4 Composition Table A binary operation on a finite set can be completely described by means of a table known as a composition table. Let be a finite set and * be a binary operation on S. Then the composition table for * is constructed in the manner indicated below. We write the elements a1, a2, ….. ,an of the set S in the top horizontal row and the left vertical column in the same order. Then we put down the element ai * aj at the intersection of the row headed by ai (1  i  n) and the column headed by to get the following table. * a1 a2 ….. ai ….. aj ….. an a1 a1 * a1 a1 * a2 ….. a1 * ai ….. a1 * aj ….. a1 * an a2 a2 * a1 a2 * a2 ….. a2 * ai ….. a2 * aj ….. a2 * an ai ai * a1 ai * a2 ….. ai * ai ….. ai * aj ….. ai * an aj aj * a1 aj * a2 ….. aj * ai ….. aj * aj ….. aj * an an an * a1 an * a2 ….. an * ai ….. an * aj ….. an * an From the composition table we infer the following results : (1) If all the entries of the table are elements of set S and each element of S appears once and only once in each row and in each column, then the operation is a binary operation. Sometimes we also say that the binary operation is well defined which means that the operation * associates each pair of elements of S to a unique element of S, i.e. S is closed under the operation *. (2) If the entries in the table are symmetric with respect to the diagonal which starts at the upper left corner of the table and terminates at the lower right corner, we say that the binary operation is commutative on S, otherwise it is said to be not commutative on S. (3) If the row headed by an element say aj, coincides with the row at the top and the column headed by aj coincides with the column on extreme left, then aj is the identity element for the binary operation * on S. (4) If each row except the topmost row or each column except the left most column contains the identity element then every element of S is invertible with respect to *. To find the inverse of an element say aj, we consider row (or column) headed by ai. Then we determine the position of identity element e in this row (or column). If e appears in the column (or row) headed by aj, then ai and aj are inverse of each other. It should be noted that the composition table is helpless to determine associativity of the binary operation. This has to be verified for each possible trial. Example: 1 Let S be a finite set containing n elements. Then the total number of binary operations on S is (a) (b) (c) (d) Solution: (c) Since a binary operation on S is a function from S × S to S, therefore the total number of binary operations on S is the total number of functions from S × S to S, which is . Example: 2 The identity element for the binary operation * defined by a * b = , (the set of all non-zero rational numbers) is (a) 1 (b) 0 (c) 2 (d) None of these Solution: (c) Let e be the identity element for the binary operation * on defined by Then, for all  for all  . Example: 3 Let z be the set of integers and o be a binary operation on z defined as for all . The inverse of an element is (a) (b) (c) (d) None of these Solution: (a) Let e be the identity element for the binary operation o defined on z given by Then for all  for all  for all  . So, 0 is the identity element for the binary operation o and z. Let x be the inverse of . Then,    Thus, is the inverse of . Example: 4 * is defined on the set of real numbers by . Then the operation * is (a) Commutative but not associative (b) Associative but not commutative (c) Neither commutative nor associative (d) Both commutative and associative Solution: (a) We have So, * is commutative on R. For any, a, b, c R, we have and So, * is not associative on R.

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