The word "binary" means composed of two pieces. A binary operation is simply a rule for combining two values to create a new value. The most widely known binary operations are those learned in elementary school: addition, subtraction, multiplication and division on various sets of numbers.


A binary operation on a set is a calculation involving two elements of the set to produce another element of the set.

Let's take a look at some creative binary operations.

bullet Situation 1:
It is possible to define "new" binary operations. Consider this example:

A new math (binary) operation, using the symbol Φ, is defined to be
a Φ b = 3a + b,
where a and b are real numbers.

1. What is 8 Φ 3 ?
Substitute the values of a and b into the right-hand side of the definition, namely 3a + b.
8 Φ 3 = 3•8 + 3 = 24 + 3 = 27

2. Is a Φ b commutative?
Does a Φ b = b Φ a for all possible values?
      3a + b = 3b + a ?     Not true for all real numbers.
If a = 8 and b = 2;     3•8 + 2 ≠ 3•2 + 8;     26 ≠ 14.
The operation Φ is not commutative for real numbers.

3. Is a Φ b associative?
Does a Φ (b Φ c) = (a Φ b) Φ c ?
      a Φ (3b + c) = (3a + b) Φ c ?    
    3a + (3b + c) = 3(3a + b) + c ?   Not true for all reals.
If a = 2, b = 3, c = 4;     3•2 + (3•3 + 4) ≠ 3(3•2 + 3) + 4;
              6 + 13 ≠ 3(9) + 4;     19 ≠ 31.
The operation Φ is not associative for real numbers.


bullet Situation 2:
Sometimes, a binary operation on a finite set (a set with a limited number of elements) is displayed in a table which shows how the operation is to be performed.

A binary operation, binaryasterisk, is defined on the set {1, 2, 3, 4}. The table at the right shows the 16 possible answers using this operation.
To read the table: read the first value from the left hand column and the second value from the top row. The answer is the intersection point.

1. What is 2binaryasterisk4 ?
2binaryasterisk4 = 2   (where the row and column intersect)

2. Is binaryasterisk commutative?
Check: 3binaryasterisk1 = 1binaryasterisk3, yes, 2 = 2.
Unfortunately, you now need to check all of the other possibilities. There is, however, a shorter way ...
draw a diagonal line from the upper left corner to the lower right corner of the table. If the table is symmetric with respect to this line, the table is commutative.
3. What is the identity element
for the operationbinaryasterisk?
Find the single element that will always return the original value. The identity element is 4. You will have found the identity element when all of the values in its row and its column are the same as the row and column headings.

4. Isbinaryasteriskassociative for these values?
 4binaryasterisk(3binaryasterisk2) = (4binaryasterisk3)binaryasterisk2
Does 4binaryasterisk(3binaryasterisk2) = (4binaryasterisk3)binaryasterisk2 ?
              4binaryasterisk(4) = (3)binaryasterisk2
                       4 = 4
  YES, this example is associative.
Unfortunately, if you were asked the general question, "Isbinaryasteriskassociative?", instead of just checking one single case as shown in #4, you would have to check ALL possible arrangements. Unlike the commutative property, there is NO shortcut for checking associativity when working with a table. But remember, it only takes one arrangement which does not work to show that associativity fails.


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