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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 03
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Recap
• Binary operation • Examples of binary operations • Motivation for defining groups• Group (Definition)
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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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Examples
• Lets look at some number sets first.
1) Is a group? Yes No
2) Is a group? Yes No
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Examples.
3) Is a group? Yes No
4) Is a group? Yes No
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Examples..
5) Is a group? Yes No
6) Is a group? Yes No
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Examples…
• Define , .
7) Is a group? Yes No
8) Is a group? Yes No
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Examples….
• We will denote by the set of positive integers, positive rational numbers and positive real numbers respectively.
9) Is a group? Yes No
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Summary
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Thank You