Properties of Binary Operations
Properties of Binary Operations: Overview
This topic covers concepts, such as, Laws of Binary Operations, Closure Property of Binary Operations, Existence of Inverse for a Binary Operation & Existence of Non-zero Divisors for a Binary Operation etc.
Important Questions on Properties of Binary Operations
Explain the inverse of binary operation.

Define a binary operation on the set as
Show that zero is the identity for this operation and each element of the set is invertible with being the inverse of

Given a non-empty set consider the binary operation : given by in , where is the power set of . Show that is the identity element for this operation and is the only invertible element in with respect to the operation

Let and be the binary operation on defined by Show that is commutative and associative. Find the identity element for on , if any.

Let be a binary operation on the set of rational numbers as Find whether the binary operation is commutative or associative or both.

Let be a binary operation on the of rational numbers,
Then find out the binary operation is

Show that division is not a binary operation on real numbers. Give one example.

Show that subtraction and division are not binary operations on .

is a binary operation, . Verify reflexive property for given binary operation.

is a binary operation, . Verify reflexive property for given binary operation.

is a binary operation, . Verify reflexive property for given binary operation.

is a binary operation, . Verify reflexive property for given binary operation.

is a binary operation, . Verify reflexive property for given binary operation.

Suppose be the multiplication operation and be the addition operation defined on . Let , then find .

Suppose be the multiplication operation and be the addition operation defined on . Let , then find .

Suppose be the multiplication operation and be the addition operation defined on . Let , then find .

Suppose be the multiplication operation and be the subtraction operation defined on . Let , then find .

Suppose be the multiplication operation and be the subtraction operation defined on . Let , then find .

Let be the binary operation defined on the set of natural numbers, where the identity element is . Find the number of invertible elements.

Let be a binary operation on , defined by . Thus is associative.
