Algebraic Operations on Sets

In Set Theory concepts and Automata Theory, finite sets are frequently used. For example, in automata theory, we use machine descriptions for finite state machines, and other automata which are represented in tuples and each element in that tuple uses several sets which are finite in nature. But to use sets we must know some of the operations involved.

In this chapter, we will go through the arithmetic operations of set theory in detail with diagrams and examples.

Union of Two Sets

The union of two sets denotes the whole set of elements that are present in A or B or both. The union of A and B is denoted as A ∪ B. For example, A = {2, 3, 4} and B = {3, 5, 6}, A ∪ B = {2, 3, 4, 5, 6} which is nothing but {x | x ∈A or x ∈B}.

In Figure-1, the first diagram represents the union operation where blue region is covering both A and B.

Intersection of Two Sets

The intersection of two sets denotes the common set which is present in both the candidate sets. The intersection of A and B is denoted as A ∩ B. For example, A = {2, 3, 4} and B = {3, 4, 6}, A ∩ B = {3, 4} which is nothing but {x | x ∈A and x ∈B}.

In Figure-1, the second diagram represents the intersection operation where common region covered by both the sets A and B is the intersection

Set Difference

The set difference of two sets denotes the set where the common elements of two sets are missing. This is little bit complex and we can understand through examples. The set difference of B from A is denoted as A - B. For example, A = {2, 3, 4} and B = {3, 4, 6}, A - B = {2} this is the removal of A ∩ B from A. This can be represented as {x | x ∈A and x ∉ B}.

In Figure-1, the third diagram represents the set difference from A to B operation where the common region covered by both the sets A and B is the intersection and from A, we remove the intersection to get the difference.

Set Complement

The complement of a set A is the all elements which are not in A, in other words the difference of universal set and the set A is the complement. We denote complement by A’ or A^c, which is a subset of a large set U, defined by A′ = {x ∈U: x ∉ A}.

In Figure-1, the fourth diagram represents the complement of set A where common region covered by A is removed from universal set U.

Cartesian Product of Two Sets

The Cartesian product of two sets A and B is denoted as A × B. For example, A = {2, 3, 4, 5} and B = {3, 4}, A × B = {(2, 3), (2, 4), (3, 3), (3, 4),(4, 3), (4, 4), (5, 3), (5, 4)}.

Power Set

The power set of a set A is the set of all possible subsets of A, such as {a, b}, and for a set of elements n, the number of elements in the power set of A is 2ⁿ. For example, for set A = {1, 2}, the power set is {{}, {1}, {2}, {1,2}} with 2² = 4 elements.

Properties of Set Operations

The following table summarizes the properties of Set Operations −

Conclusion

In this chapter, we explained the arithmetic operations on sets, including union, intersection, set difference, complement, Cartesian product and power set.

We also covered the possible properties of set theory which are very much useful while using sets in automata theory along with Boolean algebra.