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 property

There 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:

Example 1:

The inverse property is contained by the set of integers under the addition operation. This is because the integer contains an identity element 0, and if we add any number with its opposite number, then it will result as 0. Suppose there is a integer x such as x+(-x) = 0 and -x+x = 0. So we can see that when the addition operation is performed on x from the left and right sides, then it provides us with the same result as the identity 0. Hence, -x is the inverse for x in the set of integers under the addition operation.

For example:

1. 7 + (-7) = 0

Here -7 is the opposite of 7.

2. -3 + (3) = 0

Here -3 is the opposite of 3.

Example 2:

The inverse property does not contain by the natural number under the addition operation. This is because the set of natural numbers does not contain negative numbers, and we have learned that -x is the inverse for x under the addition operation. The set can only contain the inverse property if it contains the inverse in the set.

Example 3:

The inverse property does not contain by the set of integers under the operation division. This is because the identity element does not contain by the set of integers under the division operation. According to the definition of inverse, the set of integers must have the identity element for the inverse property.

Example 4:

The inverse property is contained by the set of positive integers under the multiplication operation. This is because if we multiply any number with its reciprocal number, then we will get the identity number 1. The reciprocal of any number can be determined by flipping the number. Suppose there are two integers x and y such as (a/b) * (b/a) = 1. For example: The reciprocal of 5/7 will get by flipping it like this 7/5. Similarly, the reciprocal of number 5 will get by flipping like this 1/5.

For example:

(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:

Example 1: In this example, we have a set {a, b, c} and an operation *, and we will show it with the help of an operation table.

Solution: The operation table of the set {a, b, c} under the operation * is 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.

Example 2: In this example, we have a set {a, b, c} and an operation ~, and we will show it with the help of an operation table.

Solution: The operation table of the set {a, b, c} under the operation ~ is described as follows:

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 ~.

Example 3: In this example, we have a set {x, y, z} and an operation ^, and we will show it with the help of an operation table.

Solution: The operation table of the set {x, y, z} under the operation ^ is described as follows:

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 inverse.

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 inverse.

A set can contain the inverse property under an operation only if each and every element of that set has an inverse, but in this set, elements x and z do not have an inverse.

So with the help of above operation table, we have proved that the set {x, y, z} under the operation ^ does not contain the inverse property.

Example 4: In this example, we have a set {x, y, z, v} and an operation $, and we will show it with the help of an operation table.

Solution: The operation table of the set {x, y, z, v} under the operation $ is described as follows:

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 inverse S that might exists. For this, we will take each element of the set one by one like this:

First, we will consider the element y like this:

y$y = y

So element y has the inverse because it contains the identity. As we know, the identity element is y, so element y has its own inverse.

Now we will consider the element z like this:

z$v = y

z$x = y

x$z = y

So when we consider the z$v = y, then y is not an inverse of z because v$z ≠ y.

However, z$x = y and x$z = y. Hence we can say that x is an inverse of z.

Now we will consider the element x like this:

v$x = y

x$z = y

z$x = y

So when we consider the v$x = y, then v is not an inverse of z because x$v ≠ y.

However, x$z = y and z$x = y. Hence we can say that z is an inverse of x.

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 inverse.

So we have proved that y, z, and x have an inverse S in the above set {x, y, z, v} under the operation $, but v does not have an inverse.

According to the definition of inverse property, a set can contain the inverse property under an operation only if each and every element of that set has an inverse.

So with the help of above operation table, we have proved that the set {x, y, z, v} under the operation $ does not contain the inverse property.