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Understanding Isomorphism Through Color-Coded Multiplication Tables
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Isomorphism: A First Example MAT 320 Spring 2011 Dr. Bryant
It seems like the answer is no… • Color-coding the elements of each ring shows that the multiplication tables don’t match up • However, notice something in the multiplication table for S: • This shows that 1S = 6 • Since 1 in Z5 was colored green, this means our coloring was wrong!
With this new coloring… • …we see that the two rings have exactly the same structure • When two rings have exactly the same addition and multiplication tables (under some correspondence between their elements), we say the rings are isomorphic • iso = same, morphic = structure • Finding the correspondence is the hard part!
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