## Inverse property in Discrete mathematics- If a set contains an operation in which
**each and every**element has an inverse, in this case, the set will have the**inverse property**. An element of the set will be known as the inverse element if we always get the identity element if we combine the right and left sides of the operation. We can understand this concept with the help of an example. - Suppose there is a set A, which contains two variables, x and y. Here x is used to indicate
**any arbitrary element**, and y is used to indicate the special element of the set, which is known as the inverse of x. There will also be an element e in the set, which is used to show the identity element. The symbol # is used to indicate the operation. The inverse property for the set A, with the help of operation #, is described as follows: - If there is any element x in the set, then there must be one more element y of the set in such a way that x#y = e and y#x = e.
## Note If there is a set under an operation and we want to have the inverse property, then for this, it needs to first have the identity property. Only after having the identity property, the set can have the inverse property. If the set does not contain any identity property, in this case, the definition of inverse will not make any sense. Hence, we can say that the set will not have the inverse property if it does not contain the identity property under an operation.- That means the set only will have the
**inverse property**if each and every element of a set has the identity property. - We also noticed that the left and right sides of the inverse must be the same. Suppose there are two elements, y and z, in such a way that x#y = e but y#z ≠ e and z#x = e but x#z ≠ e. In this type of case, there will be no
**inverse**. Neither y nor z will be an**inverse**because when these elements act on the x, both the right and left sides are not the same and it also does not generate**identity**as a result. - There is one more thing which we should know the inverse always comes in pairs, which means if the element x is the
**inverse**of element y, then it is compulsory that y is the**inverse**of x. There can also be one more case in which an element can have an**inverse**of itself. - The
**identity element**of the set always contains its**own inverse. For example:**Suppose there is an identity element, e. In this case, e#e = e. So on the basis of the definition, when e acts on itself on the left and right, then it will leave itself unchanged and also provide the result in the form of identity element itself.
## Note: In any set, the elements usually contain only one inverse, but there is also a possibility that an element can contain more than one inverse.## Example of Inverse propertyThere are a lot of examples to understand the inverse property with the help of using some infinite sets with operations, which we know already. These examples are described as follows:
The
1. 7 + (-7) = 0 Here -7 is the opposite of 7. 2. -3 + (3) = 0 Here -3 is the opposite of 3.
The
The
The
(5/4) * (4/5) = 1 or 20/20 = 1 7 * 1/7 = 1 or 7/7 = 1 ## Note: If we do the reciprocal of any negative number, then the sign will not be changed in the reciprocal number. For example: The reciprocal of number -3/8 will get by filliping it like this -8/3.## Examples of Inverse property using Operation table:The examples of an inverse property will become more difficult if we use the operation table to check whether the given set has an inverse property under a given operation. So the examples of inverse property for some simple set with the help of operation tables are described as follows:
As we know, 'a' is the identity element of the above set. Now we will find the inverse of the above set with the help of first finding those places where the operation on two elements generates the result as an identity element 'a'. We will do this with the help of following operation table:
With the help of above table, we have the following details: a*a = a b*b = a c*c = a So, according to this, every element has an inverse, which is described as follows: a has an inverse a b has an inverse b c has an inverse c So we can see that every element of this set has its own inverse. So with the help of above operation table, we have proved that the set {a b, c} under the operation * contains the inverse property.
As we have learned, the above table cannot contain an inverse element. Because of this, the above set will not contain the inverse property under the operation ~.
As we have learned that if we are performing any operation under the operation ^, then the set will not have the identity element. So we will try to determine the inverse S that might exist in the table. So for this, we will find the inverse of the above set with the help of first finding those places where the operation on two elements generates the result as an identity element 'y'. We will do this with the help of following operation table:
With the help of above table, we have the following details: y^y = y According to this, only element y has its own inverse. This is because, in this set, the identity element is y. But this set does not have any element a so that a^x = y or x^a = y, so we can say that x does not have an Similarly, this set does not have any element a so that a^z = y or z^a = y, so we can say that z does not have an A set can contain the So with the help of above operation table, we have proved that the set {x, y, z} under the operation ^ does not contain the
In the above table, we can see that there is an identity element y for the set {x, y, z, v} under the operation $. So we will try to determine the inverse S that might exist in the table. So for this, we will find the inverse of the above set with the help of first finding those places where the operation on two elements generates the result as an identity element 'y'. We will do this with the help of following operation table:
With the help of above table, we have the following details: x$z = y y$y = y z$v = y z$x = y v$x = y Now we will try to determine First, we will consider the y$y = y So element y has the inverse because it contains the Now we will consider the z$v = y z$x = y x$z = y So when we consider the z$v = y, then y is not an However, z$x = y and x$z = y. Hence we can say that x is an Now we will consider the v$x = y x$z = y z$x = y So when we consider the v$x = y, then v is not an However, x$z = y and z$x = y. Hence we can say that z is an Now we will consider the element v like this: z$v = y v$x = y Here z ≠ x. This is because there is no element w in such a way that w^v = y and v^w = y. Hence, we can say that v does not have an So we have proved that y, z, and x have an According to the definition of inverse property, a set can contain the So with the help of above operation table, we have proved that the set {x, y, z, v} under the operation $ does |