Difference of Sets

One of the most important and fundamental set theory operations is set difference. In addition to the difference of sets, the other set theory operations are union and intersection. The difference between two sets A and B is another set consisting of elements from A that are NOT in B.

Let’s learn more about the different types of sets, their properties, the symmetric difference of sets, Venn diagrams, and types of relation in math that are solved examples in this article.

Describe the Distinction

The process of subtracting one number from another can be viewed in a number of ways. One model that can help you understand this concept is the takeaway model of subtraction. Starting with five objects and removing two of them, the problem 5 – 2 = 3 would be demonstrated by counting the number of objects left. The difference between two sets can be found in the same way that the difference between two numbers can be found.

We’ll look at an example of a set difference as an example. To see how the difference of two sets forms a new set, consider the sets A = 1, 2, 3, 4, 5 and B = 3, 4, 5, 6, 7, 8. To find the difference A – B between these two sets, first, write all of A’s elements, then subtract any A elements that are also B elements. The set difference A – B = 1, 2 is obtained because A and B share the elements 3, 4, and 5.

Order of Set Differences

When computing the difference of sets, we must be careful about the order, just as 5 – 3 is not the same as 3 – 5. It is not commutative to compare sets. This means that if the order of the difference between the two sets is changed, the result may be different. As a result, we can conclude that A – B does not have to equal B – A for all sets A and B.

Consider the following two sets of numbers: A = 1, 2, 3, 4, 5 and B = 3, 4, 5, 6, 7, 9.

The difference A – B = 1, 2 and the difference B – A = 6, 7, 9 were discovered. As a result, we can see from this example that A – B ≠ B – A.

What’s the thing that is different Between a Collection and a Set?

The list of all the elements in set A that are not in set B is defined as the difference between two sets A and B. The set notation used to represent the difference between the two sets A and B is A B or A B. A – B are written as follows in set-builder notation:

A – B = {x / x ∈ A and x ∉ B}

The set obtained by subtracting the elements of A B from A is known as A – B.

The set obtained by subtracting the elements of A B from B is called B – A.

What Is the Best Way to Determine the Difference Between Two Sets?

To see how to calculate the difference between two sets A and B, consider the following example.

Consider the two sets A = 1, 2, 3, 4, 5 and B = 3, 4, 5, 6, 7, 9 and B = 3, 4, 5, 6, 7, 9, respectively.

Remove all elements that appear in both A and B to find the difference A – B between these two sets.

A – B = 1, 2, 3, 4, 5 – 3, 4, 5, 6, 7, 9 – 3, 4, 5, 6, 7, 9

The set of A’s remaining elements is known as A – B. As a result, A – B = 1, 2

Why is it critical to keep things in order?

We must be cautious about the order in which we compute the types of sets and their differences because the differences 4 – 7 and 7 – 4 produce different results. To use a mathematical term, we can say that the difference set operation is not commutative. This means that we can’t change the order of two sets’ differences and expect the same result in general. We can state more precisely that A – B does not equal B – A for all sets A and B.

To understand what is being talked about, consider the previous example. For the sets A = 1, 2, 3, 4, 5 and B = 3, 4, 5, 6, 7, 8, we calculated the difference A – B = 1, 2. To compare B – A, we start with B’s elements, which are 3, 4, 5, 6, 7, 8, and then remove the 3, 4, and 5 because they are shared by both B and A. The result is B – A = 6, 7, 8. This example demonstrates that A – B is not the same as B – A.

To share more:

One distinction is noteworthy enough to be given its own name and symbol. When the first set is the universal set, this is referred to as the complement, and it is used to represent the symmetric difference of sets. The expression U – A gives the complement of A. This is the collection of all universal set elements that aren’t A. We can simply state that the complement of A is the set of elements that are not members of A because it is understood that the set of elements from which we can choose is drawn from the universal set.

The complement of a set is proportional to the universal set with which we are dealing. With A = 1, 2, 3 and U = 1, 2, 3, 4, 5, the complement of A is 4, 5. If our universal set is different, such as U = -3, -2, 0, 1, 2, 3, then A’s complement is -3, -2, -1, 0. Keep an eye on which universal set is being used at all times.

Conclusion

Finally, R. C. Bose and a seminal paper published in 1939 can be credited with the systematic application of cyclic difference sets and methods for the construction of symmetric block designs. There are, however, earlier examples, such as the “Paley Difference Sets” from 1933. In 1955, R.H. Bruck is credited with extending the cyclic difference set concept to larger groups. In 1947, Marshall Hall Jr. invented multipliers.

Difference sets can be used to build a complex vector codebook that meets the difficult Welch bound on maximum cross-correlation amplitude, as discovered by Xia, Zhou, and Giannakis. The Grassmannian manifold is formed by the codebook in this form.