Binary Operations - Definition

Binary Operations - Definition

What Are Binary Operations?

• A binary operation is a rule that combines two elements to produce a new one.
• The term ‘binary’ refers to the operation involving two elements.
• These operations are commonly used in mathematics and computer science.
• Familiar examples of binary operations include addition, subtraction, multiplication, and division in arithmetic.

Key Components of Binary Operations

• Binary operations involve three parts: a set, an operation, and a result.
• The set is a collection of items (‘elements’) on which the operation can act.
• The operation is a specific process such as addition or multiplication.
• The result is the product of performing the operation on two elements from the set.

Rules of Binary Operations

• A binary operation takes two inputs and produces one output.
• The inputs must be elements from the set and the output must also belong to the same set.
• The operation must be able to be performed on any pair of elements from the set to qualify as a binary operation.
• Some operations like subtraction and division don’t always remain within the same set and therefore may not be binary operations in certain situations.

Properties of Binary Operations

• Binary operations may also have certain properties like commutativity, associativity, and the presence of identity elements and inverse elements.
• Commutativity means the order of operation doesn’t matter (e.g., a + b = b + a for addition).
• Associativity means the groupings don’t affect the final result (e.g., (a + b) + c = a + (b + c) for addition).
• An identity element is an element that when combined with any other element doesn’t change the other element (e.g., 0 for addition, 1 for multiplication).
• An inverse element for any given element is another element that combines with the given element to produce the identity element (e.g., -a is the additive inverse of a, as a + (-a) = 0).

Applicability of Binary Operations

• Understanding binary operations and their properties is crucial for investigating more complex mathematics, including abstract algebra, matrix manipulations, and computations in computer science.
• Regular practising with examples and problem-situations can significantly improve understanding of binary operations.

Remember, not only remembering definitions but also demonstrating understanding through application is required. Keep using resources such as worksheets, tutorial videos and past papers for robust revision.