Binary Operations In Mathematics: Definition, Examples, And Applications

Binary operation (on a set G)

for all a,b in G, a*b in G

In mathematics, a binary operation on a set G is a mathematical operation that combines two elements of G to produce another element of G.

Formally, a binary operation * on a set G is a function that maps each ordered pair (a, b) of elements in G to a single element c in G. We write this operation as a * b = c, where a and b are elements of G and c is the result of applying the binary operation * to a and b.

For example, addition and multiplication are binary operations on the set of real numbers, since adding or multiplying two real numbers always produces another real number. Subtraction and division are also binary operations on the set of real numbers, but only under certain conditions (e.g. the denominator in a division cannot be zero).

Other examples of binary operations include the set of integers under modulo arithmetic (i.e. addition or multiplication modulo some integer), the set of matrices under matrix multiplication, and the set of functions under function composition.

Binary operations are fundamental concepts in algebra and are used extensively in the study of groups, rings, and fields. The properties of binary operations, such as commutativity, associativity, and distributivity, play a crucial role in the development of algebraic structures and their applications in various fields of mathematics and science.

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