Binary Operations
A binary operation is basically a rule for linking two objects of a particular form, to get another object of that form. The first time you learn of binary operation is in elementary school. The objects you were making use of were numbers and the binary operations you carried out were addition, subtraction, multiplication and division.
A binary operation f(x,y) is an operation that applies to two quantities or expressions and .
A binary operation on a nonempty set A is a map f:AxA ↔ A such that
1. f is definite for every pair of elements in A, and
2. f uniquely relates each pair of elements in A to a few element of A.
Examples of binary operation on A from AxA to A include addition (+), subtraction (-), multiplication (-) and division (÷).
A binary operation on A can also be defined as a ruling that assigns to every pair of elements of A a unique element of A. Just as we are used to addition and multiplication of real numbers and these are examples of binary operations. Properly put, a binary operation is a function f: A X A → A.
For instance, addition on the integers could be defined as the function with our recognizable way of producing the answer. Therefore + (2, 3) = 5 and + (-4, 11) = 7. This appears a little too strange and we are probably happier to see 2 + 3 = 5 and – 4 + 11 = 5. Similarly, if * is a binary operation on a set then we will write a*b instead of *(a, b).
One difficulty we have is how we shall name these binary operations? When we speak about the binary operations “multiplication” and “addition” we normally think about the operations on the actual numbers. Instead of inventing new words for every new binary operation we make use of “multiplication” or “addition” but bear in mind that these signs may not mean our standard operation.
For an instance, we can describe a multiplication * on the positive integers by . Then as 3*2 = 9 we can speak of “3 times 2 is 9” (or had we named this operation addition “3 added to 2 is 9”. The context of the question ought to make it vivid. If of course we happen to be discussing three binary operations, then it would make sense to devise new names to mention every operation.
The idea of a binary operation is merely a way to generate an element of a set from a given pair of elements of the same set. For instance, in a finite set, we could list the rule in a table which we’ll call a multiplication table. (For an infinite but countable set we may be able to imply a multiplication table.).
If a set has n elements, in that case there are n2 binary operations that can be defined on that set. For our two elements set the operation! , is one of 16 binary operations possible. To observe this, a multiplication table has n rows and n columns so that there are entries to be filled. There are n ways to fill every one of the entries and therefore the total number is obtained by multiplying n by itself times.
In learning about binary operations on sets, we are more interested in those operations that have peculiar properties.
You will discover as we go along that binary operations ought not to be applied only to numbers.
A binary operation on a finite set (a set with a limited number of elements) is frequently illustrated in a table that showcases the way the operation is carried out.
This table above illustrates the operation * (“star”). The operation is functioning on the finite set A = { a, b, c, d }. The table illustrates the 16 probable calculations making use of the elements of set A.
| * | a | b | c | d |
| a | a | b | c | d |
| b | b | c | d | a |
| c | c | d | a | b |
| d | d | a | b | c |
How to read the table:
Read the first figure/letter from the left hand column and the second letter/ figure from the top row. The answer is in the cell where the row and column intersect. For instance, a * b = b, b * b = c, c * d = b, d * b = a etc.
What is the identity element for the operation * ? (What single element will always give back the original figure/letter?) The identity element is a. a * a = a, b * a = b, c * a = c, d * a = d
How to check for the Identity Element:
You will recognize 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
How to know if an operation is * commutative (Does the property x + y = y + x hold for every probable arrangements of values?) Begin to test the values: a * b = b * a is true because both sides equal b.
c * d = d * c is true because both sides equal b.
It would take a whole lot of time to be able to test every possible arrangements. There is however a simpler way to get this done. This is shown in the table below:
| * | a | b | c | d |
| a | a | b | c | d |
| b | b | c | d | a |
| c | c | d | a | b |
| d | d | a | b | c |
The operation * is commutative
Shortcut for testing for Commutativity:
It is simple to test whether an operation shown by a table is commutative. Just draw a diagonal line from upper left to lower right, and check if the table is symmetric about this line.
If the table is symmetric about the diagonal (from upper left to lower right); then the operation is commutative. It would only have taken one example of lack of commutativity for the answer for the above to be “no”
True or False:
(a * b) * c = a * (b * c) ???
Carry out the operations in the order shown by the parentheses:
(a * b) * c = a * (b * c)
b * c = a * d
d = d
So, yes, this statement is true for the table above.
This question handles only one case of the associative property for this table. Regrettably there is NO shortcut for scrutinizing associativity as there is for checking commutativity when working with a table.
If you are asked the general question if an operation possess the associative property?”, you would need to check every possible arrangements. On the other hand, if you discover one example where associativity fails, you don’t need to continue, you just have to stop and give the answer “NO” at that point.
Identities
Let A be a set on which there is a binary operation . An element e of this set is known as a left identity if for every a ε A, . In the same way, an element f is a right identity if a . f = a for every a ε A. An element which is both a right and left identity is known as an identity. (A few authors would make use of the term two sided identity for it.) For instance 0 is an identity for the normal addition on the real numbers.
Given a binary operation on a set there may be no identity element. There might be several. There might be left identities which are not right identities and vice-versa. We tend to recognize the situation in which there is a unique identity. Again, observe that an identity (left or right or both) for one operation does not have to be an identity for another operation.
Think of addition and multiplication on the reals where the identities are 0 and 1 correspondingly. This can be rather dodgy for odd binary operations on familiar sets. For instance, assuming we have a “new” binary operation on the real numbers and you are told it possesses an identity. Do NOT assume this identity is 0 or 1 (it may turn finally be be 2.7182818284590).
Another thing you must know is that being an identity is a global property in the sense that it ought to function for EVERY elements of the set. It may occur that for a few b, and and yet e is not an identity (i.e. for a few c, ).
Inverse
If identities are present for a definite binary operation then we can discuss about inverses (if they exist). If e is an identity therefore a is an inverse of b if a .b = b . a = c. Furthermore, we could define a left or right inverse (how must that be done?).
Again, we ought to point out that an element might have more than one inverse or for that matter, no inverse. Surely, if identities are not present, then it is not sensible to talk about inverses. Even if there is an identity, in that situation elements may not have an inverse.
For multiplication on the integers there is an identity, specifically 1, but 2 does not have an inverse.
In our well-known real numbers, the concept of inverse permits us to solve equations like x + 2 = 5 or 3x = 11. Nevertheless, we As well require an additional property.
Associative operations
A binary operation is said to be associative if for all elements a, b and c we have (a . b) . c = a (b . c). For handiness let’s drop the symbol for the operation and just write (ab) c = a(bc). The associative property in that case allows us to speak of abc without having to worry about whether we ought to find the answer to ab first and after that, that answer “multiplied” by c instead of calculating bc first and then “multiply” a with that answer.
In any way we handle the expression, we come up with the same element of the set. Take note that it does not mention we can do the product in any order (i.e. ab and ba may not have the same value).
Sets that possess an associative binary operation are referred to as semigroups. In a lot of practical applications of studying binary operations on sets, it is not common to find out that they are associative but it is something that cannot be assumed.
Indeed, one generally referred to as operation, the cross product on three dimensional real vectors is not associative. We ought to as well guard against thinking that associativity implies identitities and inverses exist.
Therefore, why is it that we require associativity to solve 3x=11 in the reals? Well, the inverse of 3 is 1/3 and so 1/3(3x) = 11/3 and currently we make use of associativity to modify this as (3/3) x = 11/3 etc.
Commutative operations
A binary operation is commutative if ab = ba for ALL possible a and b in the set. Addition and multiplication in the reals are commutative operations while multiplication of matrices normally is not.
Note that the definition requires ab=ba for all pairs of elements. That some element commutes with all elements does not make the operation commutative.
For a finite set whose binary operation is provided in a table, it is easy to observe whether the operation is commutative.
Distributive properties
Assuming, there are two binary operations + and * defined on a set A. We say that * distributes over + if for all a, b and c in A, a (b + c) = a . b + a . c. If we drop the circles around the plus and dot we obtain the common distributive law in the reals.
Other examples of distributive operations are set union and intersection or conjunction and disjunction in logic.
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