Editing Difference of Sets

= Difference of Sets - Definition, Explanation, and Examples =

The '''difference''' of two sets is an operation that finds elements that belong to one set but not the other. It is also called the '''relative complement'''.

== Definition of Difference ==

The difference of sets <math>A</math> and <math>B</math>, denoted by <math>A - B</math>, is the set of all elements that are in <math>A</math> but not in <math>B</math>.

Mathematically:

<math>A - B = \{ x : x \in A \text{ and } x \notin B \}</math>

== Understanding Difference ==

When we find the difference <math>A - B</math>, we look for elements that belong to set <math>A</math> only, excluding any elements that are also in <math>B</math>.

== Step-by-Step Explanation ==

1. List all elements of set <math>A</math>.  
2. List all elements of set <math>B</math>.  
3. Identify elements that are in <math>A</math> but not in <math>B</math>.  
4. Form a new set with those elements.

== Examples of Difference of Sets ==

=== Example 1: Numbers ===  
Let  
<math>A = \{1, 2, 3, 4\}</math>  
<math>B = \{3, 4, 5, 6\}</math>  

Step 1: Elements of <math>A</math>: 1, 2, 3, 4.  
Step 2: Elements of <math>B</math>: 3, 4, 5, 6.  
Step 3: Elements in <math>A</math> but not in <math>B</math>: 1, 2.  
Step 4: Difference:  
<math>A - B = \{1, 2\}</math>

=== Example 2: Letters ===  
Let  
<math>C = \{\text{a}, \text{b}, \text{c}\}</math>  
<math>D = \{\text{b}, \text{d}, \text{e}\}</math>  

Step 1: Elements of <math>C</math>: a, b, c.  
Step 2: Elements of <math>D</math>: b, d, e.  
Step 3: Elements in <math>C</math> but not in <math>D</math>: a, c.  
Step 4: Difference:  
<math>C - D = \{a, c\}</math>

=== Example 3: Students ===  
Class 1 students:  
<math>E = \{\text{John}, \text{Emma}, \text{Liam}\}</math>  
Class 2 students:  
<math>F = \{\text{Emma}, \text{Olivia}, \text{Noah}\}</math>  

Step 1: Elements of <math>E</math>: John, Emma, Liam.  
Step 2: Elements of <math>F</math>: Emma, Olivia, Noah.  
Step 3: Elements in <math>E</math> but not in <math>F</math>: John, Liam.  
Step 4: Difference:  
<math>E - F = \{\text{John}, \text{Liam}\}</math>

== Important Note ==

The difference operation is not commutative, which means:

<math>A - B \neq B - A</math> in general.

== Summary ==

* The difference of sets shows what is unique to the first set compared to the second.
* It helps in identifying exclusive elements and is widely used in data analysis and logic.

[[Category:Set Theory]]
[[Category:Set Operations]]
[[Category:Mathematics]]