{"id":7919,"date":"2022-05-11T14:28:16","date_gmt":"2022-05-11T08:58:16","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=7919"},"modified":"2025-06-02T20:45:43","modified_gmt":"2025-06-02T15:15:43","slug":"what-is-the-binary-operation","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-binary-operation\/","title":{"rendered":"What is the binary operation?"},"content":{"rendered":"\n

A binary operation<\/a> * is a set A is a function from A \u00d7 A to A. If * is a binary operation in a set A then than for the * image of the ordered pair (a, b) \u2208 A\u00d7A, we write a*b.<\/p>\n\n\n\n

For example:<\/strong> <\/p>\n\n\n\n

Addition + is a binary operation in the set of natural number N, integer Z and real number R.<\/p>\n\n\n\n

Multiplication is a binary operation in N, Q, Z, R and C.<\/p>\n\n\n\n

General properties<\/strong><\/h2>\n\n\n\n

Propery-1: <\/strong>Let A be any set. A binary operation * A \u00d7 A \u2192A is commutative if for every a, a, b \u2208A.<\/p>\n\n\n\n

a* b = b * a<\/p>\n\n\n\n

Property-2:<\/strong>Let A be a non-empty set. A binary operation *; A \u00d7 A \u2192A is associative if<\/p>\n\n\n\n

a* b) * c = a * (b * c) for every a, b, c \u2208A.<\/p>\n\n\n\n

Property-3:<\/strong> Let * be a binary operation on a non-empty set A. If there exists an element e \u2208A such that e * a = a * e = a for every a \u2208A, then we call the element e as identity with respect to * in A.<\/p>\n\n\n\n

Property-4:<\/strong> Let * be a binary operation on a non-empty set A and e be the identity element in A with respect the operation *. An element a \u2208A is invertible if there exists an element b \u2208A such that<\/p>\n\n\n\n

a* b = b * a = e<\/p>\n\n\n\n

In which case a andb are inverses of each other. For the operation * if b is the inverses of a \u2208A then we can write b =<\/p>\n\n\n\n

Cancellation laws<\/strong><\/h2>\n\n\n\n

We denote a bin. operation by * in a set A, is to satisfy.<\/p>\n\n\n\n

(i) Left cancellation law if for all a, b, c \u2208A,<\/p>\n\n\n\n

a* b = a * c \u21d2b = c<\/p>\n\n\n\n

(ii) Right cancellation law if for all a, b, c \u2208A<\/p>\n\n\n\n

b* a = c * a \u21d2b = c<\/p>\n\n\n\n

Algebraic system<\/strong><\/h2>\n\n\n\n

If A is a set and * is a bin. operation on A, then we call (A, *) an algebraic structure.<\/p>\n\n\n\n

Example: Let R be the set of real numbers, then (R, +) is an algebraic structure.<\/strong><\/p>\n\n\n\n

Example: If N denotes the set of natural numbers then (N, +) is an algebraic structure.<\/strong><\/p>\n\n\n\n

Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/p>\n\n\n\n