binary operation

The word " bi nary" 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, , 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. 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. Unfortunately, if you were asked the general question, "Is associative?", 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.