A binary operation is simply a rule for combining two objects of
a given type, to obtain another object of that type. You
first learned of binary operations in elementary school.
The objects you were using were numbers and the binary
operations you investigated were addition, subtraction,
multiplication and division. As you will discover in this
lesson, binary operations need not
be applied only to numbers.

A
binary operation on a finite set (a set with a limited
number of elements) is often displayed in a table that
demonstrates how the operation is performed. 


This table shows the operation *
("star"). The operation is working on the finite set
A = { a, b, c, d }. The table shows the 16 possible
calculations using the elements of set A.
Reading the table:
Read the first value from the left hand column and
the second value from the top row. The answer is in
the cell where the row and column intersect.
For
example, a * b = b, b * b = c, c * d = b, d * b = a and so on. 

What is the
identity element for
the operation * ?
(What single element will always return the original value?)
The identity element is
a.
a * a = a, b *
a = b, c *
a = c, d *
a = d 
Checking for the
Identity Element:
You will know the identity element when you see it,
because all of the values in its row or column are the
same as the row or column headings. 


What is the
inverse element for
b ?
(What element, when paired with
b, will return the identity element
a?)
The inverse element of b is
d.
b * d = a 


Is the operation *
commutative?
(Does the property x + y = y + x hold for ALL
possible arrangements of values?)
Start testing values:
a * b = b * a is true
since both sides equal b.
c * d = d * c is true since
both sides equal b.
WOW! Having to test ALL possible arangements could
take forever! There must be an easier way.........
The operation * is commutative.

Testing for
Commutativity Shortcut:
It is easy to check whether an operation defined by
a table is commutative. Simply draw a diagonal
line from upper left to lower right, and see if the
table is symmetric about this line. 
Since this table is
symmetric about the diagonal (from upper left to lower
right), the operation is commutative.
It would only have taken one instance
of lack of commutativity for this answer to have been
"no".


True or False:
(a * b) * c = a * (b * c) ???
Perform the operations in the order indicated by the
parentheses:
(a * b) * c = a *
(b * c)
b * c = a * d
d =
d
So, yes,
this statement is true for this table. 
This question deals with only one case of the
associative property for this table.
Unfortunately there is NO shortcut for checking associativity
as there is for checking commutativity when
working with a table.
If asked the general question "Does this operation possess the associative
property?", you would have to check ALL possible
arrangements.
On the other hand, if you find one instance where associativity fails, you are done
and the answer is "NO". 

