# MAT 112 Integers and Modern Applications for the Uninitiated

## Section13.1Definition of a Binary Operation

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 $$\otimes\text{.}$$

### Definition13.1.

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 $$b$$’).
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.

### Example13.3.Known binary operations.

1. 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 $$a+b\text{.}$$
2. 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 $$a\cdot b\text{.}$$
3. 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 $$a- b\text{.}$$
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.

### Example13.4.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 $$a,b \in T\text{:}$$
To determine the value of $$\Ty\star\Tz$$ we go to the $$\Ty$$ row which is
$$\star$$ $$\Tx$$ $$\Ty$$ $$\Tz$$
$$\cdots$$
$$\Ty$$ $$\Tx$$ $$\Ty$$ $$\Tz$$
$$\cdots$$
In the $$\Tz$$ column of this row we now find the value of $$\Ty\star\Tz\text{,}$$ namely $$\Tz\text{.}$$
When we go through all possible combinations we obtain:
\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*}
As before we use parenthesis to indicate order of operations. We first evaluate the expression in the parenthesis.

### Checkpoint13.5.Binary operation.

Let the binary operation $$\Box$$ (box) on the set $$F=\lbrace$$ n, o, p, q, r, s, t, u, v, w$$\rbrace$$ be defined by:
$$\Box$$ n o p q r s t u v w
n n n n n n n n n n n
o o q u r w v t p s n
p p v r q n p v r q n
q q r v p n q r v p n
r r p q v n r p q v n
s s p t v w r u q o n
t t r o p w q s v u n
u u v s q w p o r t n
v v q p r n v q p r n
w w n w n w n w n w n
We read o $$\Box$$ u as o box u.
Find the following.
o $$\Box$$ u =
u $$\Box$$ o =
r $$\Box$$ o =
o $$\Box$$ q =
(r $$\Box$$ o) $$\Box$$ q =
r $$\Box$$ (o $$\Box$$ q) =
Sometimes it can be useful to generate the operation table from a binary operation given by an algebraic rule.

### Example13.6.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
We read $$a\oplus b$$ as “$$a$$ mod plus $$b\text{.}$$
In Checkpoint 13.7 complete the operation table for a binary operation.

### Checkpoint13.7.Operation table.

Fill in the operation table for the binary operation $$\otimes$$ on the set $$\mathbb{Z}_{6}$$ defined by $$a \otimes b = (a \cdot b)\bmod 6$$ :
The left column represents the $$a$$ values and the top row represents the $$b$$ values.
$$\otimes$$ 1 2 3 4 5
1 5
2 4 2 4
3 3
4 2
5 4