Union of Sets

Union of Sets - Definition, Explanation, and Examples

The union of two sets is a fundamental operation in set theory. It combines all the elements from both sets into one set without repeating any element.

Definition of Union

The union of two sets $A$ and $B$ is the set containing all elements that belong to either $A$, or $B$, or both. It is denoted by:

$A\cup B$

Mathematically:

$A\cup B=\{x:x\in A\text{ or }x\in B\}$

Understanding Union

When we take the union of two sets, we gather every element that appears in either set. If an element is common to both sets, it appears only once in the union because sets do not allow duplicates.

Step-by-Step Explanation

1. Identify all elements in set $A$. 2. Identify all elements in set $B$. 3. Combine these elements into a new set. 4. Remove any duplicate elements to ensure all elements are unique.

Examples of Union of Sets

Example 1: Simple Numbers

Let $A=\{1,2,3\}$ $B=\{3,4,5\}$

Step 1: Elements of $A$ are 1, 2, and 3. Step 2: Elements of $B$ are 3, 4, and 5. Step 3: Combine all elements: 1, 2, 3, 3, 4, 5. Step 4: Remove duplicates (3 is repeated): 1, 2, 3, 4, 5.

So, $A\cup B=\{1,2,3,4,5\}$

Example 2: Letters in Words

Let $C=\{\text{a},\text{b},\text{c}\}$ $D=\{\text{b},\text{d},\text{e}\}$

Step 1: Elements of $C$: a, b, c. Step 2: Elements of $D$: b, d, e. Step 3: Combine elements: a, b, c, b, d, e. Step 4: Remove duplicates (b is repeated): a, b, c, d, e.

Thus, $C\cup D=\{a,b,c,d,e\}$

Example 3: Students in Two Classes

Class 1 students: $E=\{\text{John},\text{Emma},\text{Liam}\}$ Class 2 students: $F=\{\text{Emma},\text{Olivia},\text{Noah}\}$

Step 1: Elements of $E$: John, Emma, Liam. Step 2: Elements of $F$: Emma, Olivia, Noah. Step 3: Combine elements: John, Emma, Liam, Emma, Olivia, Noah. Step 4: Remove duplicate (Emma): John, Emma, Liam, Olivia, Noah.

Therefore, $E\cup F=\{\text{John},\text{Emma},\text{Liam},\text{Olivia},\text{Noah}\}$

Summary

• The union operation joins all unique elements from two sets.
• Duplicates are counted only once.
• It is useful in many areas like probability, logic, and computer science to combine data or possibilities.