About Symmetry About a LineDate: 06/06/2012 at 06:48:57 From: diegs Subject: Testing for commutativity I know that if the table of an operation is symmetric about a diagonal line from upper left to lower right, then the operation is commutative. But in the case of the table below, I see symmetry from lower left to upper right: * a b c a |a | b | c| b |b | c | a| c |c | a | b| Can I state this as one of the properties of a table which ensures that the given operation is commutative? Date: 06/06/2012 at 09:29:27 From: Doctor Peterson Subject: Re: Testing for commutativity Hi, Diegs. You're right that a commutative operation corresponds to a table that is symmetric about its diagonal. And this table DOES have that symmetry; the diagonal you're talking about goes from upper left to bottom right. So the symmetry (reflection) takes cells from upper right to bottom left, across that line, just as you say you see. Were you looking at the wrong diagonal, or misunderstanding what symmetry ABOUT a line means? I hope you understand WHY that symmetry represents commutativity. Commuted entries look like a*b and b*a, as marked here, for example: * a b c a | a |<b>| c | b |<b>| c | a | c | c | a | b | These are on opposite sides of the diagonal shown here: \ a b c a | \ |<b>| c | b |<b>| \ | a | c | c | a | \ | The symmetry only looks like a perfect geometrical reflection if you make sure that the cells are square, so that distances are the same horizontally and vertically: * a b c \ a b c +---+---+---+ \---+---+---+ a | a |<b>| c | a | \ |<b>| c | +---+---+---+ +---\---+---+ b |<b>| c | a | b |<b>| \ | a | +---+---+---+ +---+---\---+ c | c | a | b | c | c | a | \ | +---+---+---+ +---+---+---\ \ Does that help? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/06/2012 at 10:14:40 From: diegs Subject: Thank you (Testing for commutativity) Dr. Peterson, You are right, sir, I think I misunderstand what symmetry about a line means. Thank you so much. Hope someday I could also offer the same help to others as you do. -diegs |
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