In general, elements are the smallest irreducible parts of anything. Elements in math and science are the most basic principles or rudiments of the subject to be learned.
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An Identity element in multiplication is one that when you multiply a value by the identity element, that the original value is returned. The only identity element in multiplication is 1. If you multiply any value (other than infinity which is a special case of mathematics), the value returned will be 0. The identity element for addition is 0.
Given any elements x and y in a set and @ a binary operator, x @ y is also an element of the set.
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Pure Mathematics is the branch of mathematics that deals only with mathematics and how it works - it is the HOW of mathematics. It is abstracted from the real world and provides the "tool box" of mathematics; it includes things like calculus. Applied mathematics is the branch of mathematics which applies the techniques of Pure Mathematics to the real world - it is the WHERE of mathematics; it includes things like mechanics. Pure Mathematics teaches you HOW to integrate, Applied mathematics teaches you WHERE to use integration.
The law of closure states that a set of numbers is closed under an operation if when the operation is performed on any two elements of the set the result is an element of the set