Editing Intersection of Sets

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

The '''intersection''' of two sets is an important set operation that finds all elements common to both sets.

== Definition of Intersection ==

The intersection of two sets <math>A</math> and <math>B</math> is the set containing all elements that are in both <math>A</math> and <math>B</math>. It is denoted by:

<math>A \cap B</math>

Mathematically:

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

== Understanding Intersection ==

When we take the intersection of two sets, we look for elements that appear in both sets simultaneously.

== Step-by-Step Explanation ==

1. Identify all elements of set <math>A</math>.  
2. Identify all elements of set <math>B</math>.  
3. Find the elements that are present in both <math>A</math> and <math>B</math>.  
4. Form a new set with these common elements.

== Examples of Intersection 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: Common elements: 3, 4.  
Step 4: Intersection:  
<math>A \cap B = \{3, 4\}</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: Common elements: b.  
Step 4: Intersection:  
<math>C \cap D = \{b\}</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: Common element: Emma.  
Step 4: Intersection:  
<math>E \cap F = \{\text{Emma}\}</math>

== Summary ==

* The intersection of two sets contains only the elements common to both.
* If there are no common elements, the intersection is the empty set <math>\emptyset</math>.
* The intersection helps in finding shared characteristics or common data points.

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