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Lesson 104: Working with Binary Operations
A binary operation is an operation that works with two numbers at a time. In this lesson we'll explore the topic, and learn some new definitions.
Addition and subtraction are examples of binary operations. All that means is that those operations work on two numbers at a time. Even if you're adding up a whole list of numbers, you're really only adding two at a time.
We are often asked to determine if a particular operation is closed. If an operation is closed, what it means is that if you perform that operation on a given set of numbers, your answer will always be a member of that set. For example, the set of integers is closed under addition. That means that if we add any two integers, our answer will be an integer. Is the set of integers closed under division? What if we do 1 / 2? Our answer is a fraction (or decimal). That is not an integer, so that means that it it's not closed.
Very often an exam question will make up a fictitious operation for a small set of numbers, and they will show you a table so that you can see how the operation works. We've worked with tables before, such as multiplication tables. They just help us find the answer to whatever two numbers we are working with. The same is true of a table that will be provided in the question.
Here is an example. Look at the table below:
| ♥ | L | O | V | E |
| L | E | L | O | V |
| O | L | O | V | E |
| V | O | V | E | L |
| E | V | E | L | O |
This table defines an operation called ♥ for the set {L, O, V, E}. This may seem very strange. We've never worked with the ♥ symbol in math. That's OK. This is a just an imaginary operation that has been defined for a set of four imaginary elements. Let's try to use the table. What is V ♥ E? Using the table we can see that it is L. Just find the V down the left hand column, and the E along the top, and see what letter appears where they intersect. This is just how you used a multiplication table.
Here are the typical questions that you would be asked. Is the set {L, O, V, E} closed under the operation ♥? The answer is yes. No matter which two letters we work with, our answer is always a member of the set. For example, we'll never get G as an answer.
Another question would be for you to state the identity element. Remember that the identity element is the element that always gets us back to the original value. For example, in addition, the identity element is 0. If you add 0 to any number, the answer is the number that you started with. In multiplication, it is 1. n x 1 = n.
What is the identity element in the set shown in the table, under the operation shown? The answer is the letter O. You can see this by looking across the O row. The results in that row in the same order as the letters along the top. Another way to check this is that any element that we perform the ♥ operation with O, gives us back the original element. For example L ♥ O = L. V ♥ O = V.
What is the inverse of V? What this is asking is what element we have to do the ♥ operation on with V, in order to get our identity element as the answer. Our identity element is O. The answer to the question is L. When we do V ♥ L, we get O, our identity element. What is the inverse of E? If you look in the table, you'll see that the inverse of E happens to be E.
Just to help explain the concept, let's look at inverses in operations that we are used to. What is the inverse of 5 in addition? It's is -5. That's what we have to add to 5 in order to get our identity element of 0. What is the inverse of 7 in multiplication? It is 1/7. That's what we have to multiply 7 by to get 1, the identity element in multiplication.
You'll have more practice with this later. Be sure to memorize the terms presented.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.