Definition 13.1.1.
A binary operation \(\bullet\) on a set \(S\) is a function \(\bullet:S\times S\to S\text{.}\) For the image of \((a,b)\in S\times S\) under the function \(\bullet\) we write \(a\bullet b\) (read ‘\(a\) dot \(b\)’).
Section 13.1 Definition of a Binary Operation
A binary operation is a function. Its input is two elements of the same set \(S\text{,}\) and its output is also an element of \(S\text{.}\) We can write two elements \(a\) and \(b\) of \(S\) as the pair \((a,b)\text{.}\) Since \((a,b)\) is an element of the Cartesian product \(S\times S\text{,}\) we describe a binary operation as a function from \(S\times S\) to \(S\text{.}\)
We use symbols for binary operations instead of function names, just as we do for addition and multiplication of integers. Addition uses the symbol \(+\text{,}\) and multiplication uses the symbol \(\cdot\text{.}\) We use symbols such as \(\star\) and \(\bullet\) for arbitrary (non-specific) binary operations. We also define new binary operations using the symbols \(\oplus\) and \(\otimes\text{.}\)
The video in Figure 13.1.2 gives an overview of the rest of the section.
We give examples of binary operations that we saw earlier.
Example 13.1.3. Known binary operations.
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The addition of integers \(+:\Z\times\Z\to\Z\) is a binary operation on \(\Z\text{.}\) We denote the image of \((a, b) \in \Z \times \Z\) by \(a+b\text{.}\)
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The multiplication of natural numbers \(\cdot:\N\times\N\to\N\) is a binary operation on \(\N\text{.}\) We denote the image of \((a, b) \in \N \times \N\) by \(a\cdot b\text{.}\)
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The subtraction of integers \(-:\Z\times\Z\to\Z\) is a binary operation on \(\Z\text{.}\) We denote the image of \((a, b) \in \Z \times \Z\) by \(a- b\text{.}\)
Like other functions, a binary operation can be described in several ways. If the set is small, we sometimes use a table to describe the binary operation.
Example 13.1.4. A binary operation given by a table.
Let \(T:=\{\Tx,\Ty,\Tz\}\text{.}\) The binary operation \(\star:T\times T \to T\) is given by the operation table:
| \(\star\) | \(\Tx\) | \(\Ty\) | \(\Tz\) |
|---|---|---|---|
| \(\Tx\) | \(\Tz\) | \(\Tx\) | \(\Ty\) |
| \(\Ty\) | \(\Tx\) | \(\Ty\) | \(\Tz\) |
| \(\Tz\) | \(\Ty\) | \(\Tz\) | \(\Tx\) |
From the table, we can obtain \(a\star b\) (read “\(a\) star \(b\)”) for each \(a,b \in T\text{:}\)
| \(\star\) | \(\Tx\) | \(\Ty\) | \(\Tz\) |
|---|---|---|---|
| \(\cdots\) | |||
| \(\Ty\) | \(\Tx\) | \(\Ty\) | \(\Tz\) |
| \(\cdots\) | |||
In the \(\Tz\) column of this row now we find the value of \(\Ty\star\Tz\text{,}\) namely \(\Tz\text{.}\)
When we go through all possible combinations we get:
\begin{gather*}
\Tx \star \Tx=\Tz\\
\Tx\star \Ty=\Tx\\
\Tx\star \Tz=\Ty\\
\Ty \star \Tx=\Tx\\
\Ty\star \Ty=\Ty\\
\Ty\star \Tz=\Tz\\
\Tz \star \Tx=\Ty\\
\Tz\star \Ty=\Tz\\
\Tz\star \Tz=\Tx
\end{gather*}
We can use an algebraic rule for a binary operation to build its operation table.
Example 13.1.5. The binary operation \(\oplus:\Z_5\times\Z_5\to\Z_5\).
The operation table for the binary operation \(\oplus:\Z_5\times\Z_5\to\Z_5\) given by \(a\oplus b=(a+b)\fmod 5\) is:
| \(\oplus\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | 1 |
| 3 | 3 | 4 | 0 | 1 | 2 |
| 4 | 4 | 0 | 1 | 2 | 3 |
