Integers and Modern Applications for the Uninitiated

A binary operation can be considered as a function whose input is two elements of the same set \(S\) and whose output also is an element of \(S\text{.}\) Two elements \(a\) and \(b\) of \(S\) can be written as a pair \((a,b)\text{.}\) As \((a,b)\) is an element of the Cartesian product \(S\times S\) we specify a binary operation as a function from \(S\times S\) to \(S\text{.}\)

We use symbols to represent binary operations instead of function names, just as we do with addition and multiplication of integers. Addition uses the symbol \(+\) and multiplication uses the symbol \(\cdot\text{.}\) We will use symbols such as \(\star\) and \(\bullet\) to represent arbitrary (non-specific) binary operations, and we will also define new binary operations using the symbols \(\oplus\) and \(\otimes\text{.}\)

We give an overview over the remainder of the section in the video in Figure 13.2.

We give examples for binary operations that we have encountered before.

As is the case for other functions, there are several ways of specifying a binary operation. If the set is small, we sometimes specify the binary operation by a table.

As before we use parenthesis to indicate order of operations. We first evaluate the expression in the parenthesis.

Sometimes it can be useful to generate the operation table from a binary operation given by an algebraic rule.

In Checkpoint 13.7 complete the operation table for a binary operation.