Union of Sets

1.0What is the Union of Sets? 

The union of sets is a binary operation that combines all the elements from two or more given sets into a single new set. The new set contains every element that is present in at least one of the original sets. It's like merging two groups of people to form a single, larger group.

2.0Union of Sets Definition

Formally, the union of two sets, say set A and set B, is a set that consists of all the elements that belong to set A or set B or both. This is an inclusive "or."

• Union of sets symbol: The symbol used to denote the union of sets is '∪'. So, the union of sets A and B is written as A∪B.
• Set-Builder Notation: A∪B={x∣x∈A or x∈B}

This notation reads: "A union B is the set of all elements x such that x is an element of A or x is an element of B."

3.0Union of Sets Symbol

The union of sets symbol is “($\cup$)”, which is a stylized letter “U” (not to be confused with the English alphabet U). This symbol is universally accepted in set theory and mathematics.

Example of Usage: If ( A = {1, 2, 3} ) and ( B = {3, 4, 5} ), then ( A $\cup$ B ) denotes their union.

4.0Union of Sets Formula

The union of sets formula is helpful not just for writing the union but also for calculating the number of elements (cardinality) in the union, especially if the sets overlap.

Formula for Two Sets

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Where:

• ( $n(A\cup B)$ ) = Number of elements in the union
• ( n(A) ) = Number of elements in set A
• ( n(B) ) = Number of elements in set B
• ( $n(A\cap B)$ ) = Number of elements common to both sets (intersection)

Formula for Three Sets

$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(A\cap C)+n(A\cap B\cap C)$

This formula is an application of the principle of inclusion-exclusion and is critical in solving JEE problems with overlapping sets.

5.0Union of Sets Example

Let’s understand with a practical example:

Example 1: Let ( A = {2, 4, 6, 8} ) and ( B = {6, 8, 10, 12} ).

• Step 1: List all elements of A and B.
• Step 2: Combine all elements, removing duplicates.

$A\cup B=2,4,6,8,10,12$

Here, 6 and 8 appear in both sets but are listed only once in the union.

Example 2: If ( $X=\text{a, b, c}$ ) and ( $Y=\text{b, c, d, e}$ ), then:

$X\cup Y=\text{a, b, c, d, e}$

6.0Union of Sets Venn Diagram

A Venn diagram is an excellent visual tool to understand set operations. For the union of two sets, A and B, the union of sets diagram shows the combined area of both circles representing sets A and B.

In the above Venn diagram, the red-coloured portion represents the union of both sets A and B.

Thus, the union of two sets A and B is given by a set C, which is also a subset of the universal set U such that C consists of all those elements or members which are either in set A or set B or in both A and B i.e.,

C = A ∪ B = {x : x ∈ A or x ∈ B}

7.0Union of Sets Diagram Representation

Apart from Venn diagrams, sets can be represented using roster form, set-builder notation, or tabular form.

Roster Form Example

If ( A = {1, 3, 5, 7} ) and ( B = {5, 7, 9} ), then: $A\cup B=1,3,5,7,9$

Set-Builder Notation Example

$A=x:x\text{ is an odd integer less than 8}$

$B=x:x\text{ is an odd integer greater than 4 and less than 10}$

$A\cup B=x:x\text{ is an odd integer less than 10}$

8.0Properties of Union of Sets

Understanding the properties of union of sets is crucial for problem-solving and proofs in JEE Mathematics.

1. Commutative Law : $A\cup B=B\cup A$

The order of union does not affect the result.

2. Associative Law: $(A\cup B)\cup C=A\cup (B\cup C)$

• Grouping of sets does not change the outcome.

3. Identity Law: $A\cup \varnothing =A$

• The union of any set with the empty set is the set itself.

4. Idempotent Law: $A\cup A=A$

• Union of a set with itself gives the same set.

5. Domination Law: $A\cup U=U$, where ( U ) is the universal set.

6. Subset Law : If ( $A\subseteq B$ ), then ( $A\cup B=B$ ).

7. Distributive Law $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$

8. De Morgan’s Law: $(A\cup B{)}^{\prime }=A^{\prime }\cap B^{\prime }$, where the prime denotes the complement of a set.