Mathematical Operations

==2024-11-18

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Theory

Mathematical Operations are the functions that we can perform among the members of the set.

What we operate on is called Operand. What we operate is called Operator.

The process of operator applying certain function on the operands is called an operation.

Definition

For a given set A, an operation (*) is defined as the function taken from $A\times A$ to $A$ and is well-defined such that for every $(a,b)\in A^2$, $a\ast b\in A$.

i.e $\ast :A\times A\to A$

Important

The above definition is specially for binary operations.

There are many Types of Mathematical Operations.

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Examples

Addition between two numbers is a mathematical operation.

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Q&A

How can we know if Mathematical Operations are well defined?

If the outcome of the operation lies in the set from which we take the operands then the operation is well defined. i.e for every $(a,b)\in A\times A$, $a\ast b\in A$.

Important

If an operation is well defined, we generally talk and discuss about the Mathematical Properties of the operation. But if it is not well defined, we do not consider the properties but look for a different operation all together.

Example

You cannot divide Matrices together. So we look for another operation called Inverse of a Matrix. And then perform Multiplication of Matrices on those matrices.

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PTR