Union in Mathematics

Introduction

One of the set operations used in set theory is the union of sets. Other set operations include difference and intersection, in addition to set union. A unique operator is used to represent all set operations. The arithmetic addition of sets is equivalent to the union of sets. The set that contains all of the elements in both sets is the union of two given sets. ‘∪’’ is the sign denoting the union of sets.

Let’s consider an example for the two sets, A and B. the union, A ∪ B (read as A union B) lists all the elements of set A as well as set B. Thus, for these predefined two given sets, Set A = {1,2,3,4,5} and Set B = {3,4,6,8}, A ∪ B = {1,2,3,4,5,6,8}.

Notation of Union of Sets

Each set action is represented by a distinct mathematical notation. The mathematical notation for representing the union of two sets is ‘∪’. The operands surrounding this operator are known as infix notation.

Consider two sets P and Q, where P = {2,5,7,8} and Q = {1,4,5,7,9}. P ∪ Q = {1,2,4,5,7,8,9}.

Venn Diagram of Union of Sets

Venn Diagrams refer to the diagrams that are used to depict or explain the relationship between the given set of operations. A Venn diagram can be used to represent any set operation. Each set is represented by a circle in a Venn diagram. Although the union operation between two sets was utilized in this example, the Venn diagram is frequently used to depict the union of several sets, as long as the sets are finite.

Properties of the Union of Sets

Let’s learn about some of the essential properties of the union of sets in this section. When executing a set union, it is critical to take these qualities into account.

Commutative Property

The resultant set is unaffected by the order of the operating sets because of the union’s commutative characteristic. This means that if the operands’ positions are modified, the answer remains unchanged and is unaffected. We can express this mathematically as A ∪ B = B ∪ A

Property of Association

When the sets are grouped using parentheses, the result is unaffected by the associative attribute of the union. This means that if the location of the parentheses is modified in any set expression that involves union, the resultant set is unaffected.

Idempotent Property

The idempotent property states that any set can be combined with any other set to produce the same set. It can be shown mathematically as A ∪ A is equal to A.

Let’s prove this for A is equal to {2,4,6,8,10}

Thus, A ∪ A is equal to {2,4,6,8,10} ∪ {2,4,6,8,10} is equal to {2,4,6,8,10} is equivalent to A.

Property of Ⲫ/ Identity Law

The union of any set with a null set or an empty set results in the set itself, according to the property of a null set. We can write it as A ∪ Ⲫ = A in mathematics.

Let’s prove this for A = {p,q,r}

Thus, A∪∅ = {p,q,r} ∪ {} = {p,q,r}

Property of Universal Set

The union of the universal set with any set results in the universal set, according to the universal set’s property. We can write it as A ∪ U = U in mathematics.

Let’s prove this for A is equal to {a,e} and U is equal to {a,b,c,d,e,f,g,h}

then, A∪U = {a,e} ∪ {a,b,c,d,e,f,g,h} = {a,b,c,d,e,f,g,h} = U

Important Notes on Union of Sets 

• When two sets are combined, a whole new set is created that has all of the items from both sets.
• All components that are present in the first set, the second set, or both sets are included in the resultant set.
• When two disjoint sets are combined, a new set is created that contains elements from both sets.
• The resultant set is unaffected by the order of the operating sets because of the union’s commutative characteristic.
• To determine the cardinal number of the union of sets, it is important to use the formula: n(A ∪ B) is equal to n(A) + n(B) – n(A ∩ B).

Conclusion

One of the set operations used in set theory is the union of sets. Other set operations include difference and intersection, in addition, to set union. A unique operator is used to represent all set operations. The arithmetic addition of sets is equivalent to the union of sets. The set that contains all the elements in both sets is the union of two given sets.