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?
On
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.
On
A 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 ??
On
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 $*$ 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.