Binary Operations — Chapter Summary
ICSE · Class 12 · Mathematics
Summary of Binary Operations for ICSE Class 12 Mathematics. Part of the ICSE Class 12 Mathematics syllabus. Binary operation is the basic idea.
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Overview
Binary operation is the basic idea of combining two elements of a set to produce one element of the same set. The chapter studies when an operation is valid on a set, and then examines its properties such as commutativity, associativity, identity element, inverse, distributive law, and operation tab
Key Concepts
A binary operation on a non
A binary operation on a non-empty set A is a function f: A × A → A. By convention, f(a, b) is written as a f b, or simply a * b when the operation is
A set A satisfies closure law
A set A satisfies closure law with respect to an operation * if a * b belongs to A for every a, b in A. If this happens, * is a binary operation on A
A binary operation * is commutative
A binary operation * is commutative if a * b = b * a for every a, b in A. A set with this property is called Abelian with respect to the operation.
A binary operation * is associative
A binary operation * is associative if (a * b) * c = a * (b * c) for every a, b, c in A.
An element e in A
An element e in A is an identity element of * if a * e = e * a = a for every a in A. If it exists, it is always unique.
Learning Objectives
- Identify whether a given operation is a binary operation on a set.
- Use closure law to test whether an operation maps every ordered pair from A × A back into A.
- Check commutative law and associative law for a binary operation.
- Find the identity element, if it exists, and verify its uniqueness.
- Find invertible elements and their inverses, if they exist, and use uniqueness of inverse.
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