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MTH376: Algebra

For

 Master of Mathematics

By

Dr. M. Fazeel AnwarAssistant Professor

Department of Mathematics, CIIT Islamabad

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Lecture 02

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 Recap

• Set• Some important number sets• Function

A function is a rule that assigns to each element in a unique element in . Mathematically is a function if

i. for all

ii. implies .

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Binary Operation

• A binary operation on a set is a rule which assigns to every ordered pair of elements of an element is

• More precisely is a binary operation if

is a function.

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Examples

• Let denotes the operation of addition on the set of integers Then is a binary operation. Similarly addition is a binary operation on  and 

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Examples(continued…)

• Let denotes the operation of multiplication on the set of integers Then is a binary operation. Similarly multiplication is a binary operation on  and 

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More examples

• Let be the set of real valued functions defined for all real numbers, then the usual sum and product of functions are binary operations on

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Some non examples

• Subtraction is not a binary operation on Also division is not a binary operation of .

• The operation defined by is not a binary operation. Also is not a binary operation.

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Motivation for defining groups

Solution of linear equations

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Motivation (continued…)

Solve the following equation:

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Group

• A group  is a set together with a binary operation on such that the following axioms are satisfied:

1. The binary operation is associative.

2. There is an element such that for all

3. For each there is an element such that  for all

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Discussion and remarks

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Summary

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Thank You