How to Determine if a Mathematical System is Commutative (Table Method)
Obtain the mathematical system table you need to interpret., Look down the length of the downward sloping diagonal., Find the reflective objects across each of the table's diagonals., Look for symmetry in the table.
Step-by-Step Guide
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Step 1: Obtain the mathematical system table you need to interpret.
It'll work for all mathematical systems, but for you to understand, think about a set of multiplication tables from your childhood.
Organize the table so that a*b is the value in a's row and b's column.
Also make sure that the numbers written down the leftmost column appear in the same order as the ones reading across the top row. -
Step 2: Look down the length of the downward sloping diagonal.
These don't matter for commutativity because a*a always equals a*a (in the reverse order). , For every entry not on the main diagonal, find its reflection across that diagonal.
This pair of entries correspond to a*b and b*a for some arguments a and b.
If the operation is commutative, these must be equal.
Equivalently, the table entries must be symmetric under reflection across the main diagonal., If the table is the required symmetry, then it describes a commutative operation.
If any entry does not match its opposite across the diagonal, than those table entries represent a counterexample where a*b does not equal b*a and the operation is not commutative. -
Step 3: Find the reflective objects across each of the table's diagonals.
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Step 4: Look for symmetry in the table.
Detailed Guide
It'll work for all mathematical systems, but for you to understand, think about a set of multiplication tables from your childhood.
Organize the table so that a*b is the value in a's row and b's column.
Also make sure that the numbers written down the leftmost column appear in the same order as the ones reading across the top row.
These don't matter for commutativity because a*a always equals a*a (in the reverse order). , For every entry not on the main diagonal, find its reflection across that diagonal.
This pair of entries correspond to a*b and b*a for some arguments a and b.
If the operation is commutative, these must be equal.
Equivalently, the table entries must be symmetric under reflection across the main diagonal., If the table is the required symmetry, then it describes a commutative operation.
If any entry does not match its opposite across the diagonal, than those table entries represent a counterexample where a*b does not equal b*a and the operation is not commutative.
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Tyler Green
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