Trying to understand binary operations, but seriously confused. I was looking at all the videos on youtube, forums, but I think I must be missing something.
I have a set $$S = \{a, b, c, d, e \}$$ and for $x,y \in S$ define the binary operation $*$ on $S$ by $x*y=y$. It is this form, so please stop changing it!
I'm trying to build a table to represent it, to show $*$ on $S$.
Any suggestions?
A binary operation $*$ on a set $S$ is nothing more than a bunch of assignments. Here, we assign to each pair of elements of $S$ another element of $S$. So, either $*$ is defined for you already, or you define $*$ so that
$a*a = $ something in $S$
$a*b = $ something in $S$ (could be the same something as above, or it could be different)
$a*c = $ something in $S$ (ditto)
and so on for every pair of elements.
You mentioned an operation table, which is a fine way to represent an operation on a set of this size. Draw a $6 \times 6$ grid similar to the one in this question, only instead of using the symbols in that grid, use the symbols $a$, $b$, $c$, $d$, and $e$. The row in yellow is usually called the master row, and you should have each element of $S$ in it. Similarly for the master column. Now, for the "lower-right" $25$ elements, fill in each box with one of the elements of $S$.
Now, to find, say, $b*d$, start with the row where the leftmost (yellow) entry is $b$. Search along that row until you find the column where the topmost (yellow) entry is $d$. The value in that box is what you have assigned to $b*d$.
Edit: Now that I see that for each $x,y \in S$, $x*y=y$, this task becomes clearer. Simply assign $a*a = a$, $a*b = b$, and so forth.
Well stating that $x*y = y$ pretty well defines it in my opinion...if you mean list the full function (i.e. each of $25$ permutations of $x, y$), then you just list it.
As a table:
\begin{array}{c | c c c c c} * & \textbf{a} & \textbf{b} & \textbf{c} & \textbf{d} & \textbf{e} \\ \hline \textbf{a} & a & b & c & d & e \\ \textbf{b} & a & b & c & d & e \\ \textbf{c} & a & b & c & d & e \\ \textbf{d} & a & b & c & d & e \\ \textbf{e} & a & b & c & d & e \\ \end{array}
Each row is identical because any value $*a$ gives $a$, and any value $*b$ gives $b$, etc.
A binary operation is a function $f(x,y)$ that produces a third element that may or maynot be distinct from the previous two elements, it is however more commonly written with some figure like $x\cdot y$, $x\circ y$, $x\diamond y$ etc.
In many instances when we have that $\forall x,y\in S$ and $x\circ y\in S$ we say it is a closed operation because it produces nothing outside of the given set, in many instances we also have what is called the neutral/identity element where $x\circ e=x$ for all $x\in S$.
For representing it with a small finite set, like your $S$ you can do it with a table like on here where the top row represent either the left or right element while the first column represent the other element and then you see what they produce.
$x\circ y = a$ for any $x,y\in S$ if you want and it goes.
– Zelos Malum May 29 '15 at 04:36A binary operation is an operation that takes two elements (pair) of some set and then map them uniquely to an element in the same set.
Formally, a binary operation on a non empty set $S$ is a map $$f:S \times S \to S$$ that have the following properties
(1) $f$ is defined for every pair of elements in $S$
(2) $f$ uniquely associates each pair of elements in $S$ to some element of $S$
You can have a lot of binary operations on the set $S$ that you provided but for instance if your binary operation $*$ has map $a*b = c$ then it can't be true that $a*d = c$ as well, it fails the second property of the binary operation. This is how you can build your table, assume $a*b=c$ then $a*c \neq c$ but maybe $=d,e,f$ and if you chose that $a*c =d$ then $a*d \neq c,d$ but it can be equal to $e,f$ and so on, you see where is that going !
Below is a possible table for the binary operation $*$ on $S$
$$ \begin{array}{c|lcr} *& e & a & b & c & d \\ \hline e & e & a & b & c & d \\ a & a & b & c & d & e \\ b & b & c & d & e & a \\ c & c & d & e & a & b \\ d & d & e & a & b & c \end{array}$$
is an example of a table for a binary operation that satisfies the two properties, as a matter of fact, this is a table of a group with operation $*$ as well as the identity is the element $e$, This table is actually very easy to build, do you see the shifting in each row and also this table is symmetric meaning that row 1= col1, row 2= col2 and so on, It's also commutative table, $a*b = b*a = e$ and so on.
Keep in mind, that this is not the only table that you can build, actually a table that has distinct elements row wise and column wise is also valid, for example a row which has $$a \space a \space b \space c \space d$$ is not allowed but a row which has $$c \space d \space e \space b \space a$$ is allowed , same for columns, there are many tables that can be generated, can you count how many tables are possible ??
