Properties and tables

An operation can have some useful properties. Each one is tested with the operation’s rule or read from its table. A table lists the elements along the top and down the side, and the entry in row $x$, column $y$ is $x * y$. The tables below use $\otimes$, multiplication modulo 5 on $\{1, 2, 3, 4\}$ (see modular arithmetic↺).

Closure

An operation is closed on a set if $a * b$ is always in the set, whatever $a$ and $b$ you pick from it. In a table, every entry must be one of the elements.

Commutative

An operation is commutative if $a * b = b * a$ for every $a$ and $b$: the order doesn’t matter. To test a rule, swap $a$ and $b$ and see if you get the same expression. In a table, every entry must match its mirror image across the leading diagonal.

Associative

An operation is associative if $(a * b) * c = a * (b * c)$ for every $a$, $b$ and $c$: it doesn’t matter where the brackets go.

The identity element

The identity $e$ leaves every element unchanged: $a * e = e * a = a$ for every $a$. For ordinary addition it is 0, and for multiplication it is 1.

To find it from a rule, write out $a * e = a$ and solve for $e$. Then check that $e * a = a$ as well. In a table, the identity’s row and its column both repeat the headings.

Inverses

The inverse of $a$, written $a^{-1}$, is the element that combines with $a$ to give the identity: $a * a^{-1} = a^{-1} * a = e$. Find the identity first. From a rule, solve $a * x = e$. In a table, go along row $a$ until you find $e$: its column is the inverse.

Try it

Pick a table, then a property. Look for the tables that fail: one has no identity, one is not commutative, and one is not closed.

Building and using a table

To build a table from a rule, work out each entry with the rule. If the question says modulo $n$, reduce each answer to its remainder on dividing by $n$. To solve an equation such as $3 \otimes n = 2$ from the table, go along row 3, find the 2, and read off its column. On the diagonal you find $n \otimes n$.

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