Binary operations

Basic Understanding

A binary operation is a calculation that combines two elements to produce another element.
Examples of common binary operations include addition, subtraction, multiplication and division.
In the context of sets, a binary operation associates every pair of elements in a set with another element in the same set.

Properties

Binary operations can either be commutative or non-commutative. In a commutative binary operation, the order of the elements does not affect the result (like addition or multiplication), while in a non-commutative one, the order of the elements does affect the result (like subtraction or division).
If there exists an element in the set that, when combined with any other element via the binary operation, results in the same other element, this is called an identity element. For example, zero is the identity element for addition, and one is the identity element for multiplication.
A binary operation might have inverse elements. An element is an inverse of another if, when combined via the operation, they produce the identity element. For example, the inverse of a number x under addition is -x, because x + (-x) = 0.

Associative and Distributive Laws

A binary operation is associative if the way elements are grouped doesn’t affect the result. For instance, in addition, (a+b) + c equals a + (b+c).
A set with two binary operations might obey the distributive law. For example, multiplication distributes over addition, because a(b+c) equals (ab) + (a*c).

Binary Operations on Algebraic Structures

Binary operations can be performed on different algebraic structures, including groups, rings and fields.
In a group, there is one binary operation, an identity element, and every element has an inverse. In a ring, there are two binary operations, and they form a group under addition. In a field, both addition and multiplication are binary operations, and they form a group separately.

Remember to grasp these concepts by going through illustrative examples and solving various problems. Make use of available past papers and look for online tutorials to strengthen learning and solve any conceptual challenges.