Editing Equal Sets

= Equal Sets - Definition and Examples=

In set theory, '''equal sets''' are sets that contain the '''exact same elements'''. The order of elements or how they are written does not matter, only the content does.

== Definition of Equal Sets ==

Two sets A and B are said to be '''equal''' if they have '''exactly the same elements'''. This means every element of set A is in set B, and every element of set B is in set A.

* Mathematically:
<math>A = B \iff (x \in A \Rightarrow x \in B) \text{ and } (x \in B \Rightarrow x \in A)</math>

== Properties of Equal Sets ==

* Both sets must have the '''same number of elements'''.
* Each element in one set must be present in the '''other set'''.
* The '''order''' of elements and '''repetition''' do not matter.
* If <math>A = B</math>, then <math>B = A</math> ('''symmetry''').

== Symbolic Representation ==

If <math>A</math> and <math>B</math> are equal sets, we write:
<math>A = B</math>

== Examples of Equal Sets ==

=== Example 1: ===
<math>A = \{1, 2, 3\}</math>  
<math>B = \{3, 2, 1\}</math>  
'''A = B''' because they have the same elements, even though the order is different.

=== Example 2: ===
<math>C = \{a, b, c\}</math>  
<math>D = \{a, a, b, c\}</math>  
'''C = D''' because repetition does not change the set. D still contains only a, b, and c.

=== Example 3: ===
Let E be the set of vowels in the word "APPLE".  
The vowels are: A and E (note: repeated letters are not included more than once).

<math>E = \{A, E\}</math>

Let F be another set:  
<math>F = \{E, A\}</math>

Then '''E = F''' because both sets have the same elements, even though the order is different.

== How to Verify if Sets are Equal ==

1. List all distinct elements of both sets.
2. Compare if both sets contain the exact same elements.
3. Confirm that:
   <math>A \subseteq B</math> and <math>B \subseteq A</math>  
   If both are true, then <math>A = B</math>.

== Difference Between Equal and Equivalent Sets ==

{| class="wikitable"
! Property !! Equal Sets !! Equivalent Sets
|-
| Definition || Same elements || Same number of elements
|-
| Symbol || <math>A = B</math> || <math>A \approx B</math>
|-
| Example || <math>\{1, 2, 3\} = \{3, 2, 1\}</math> || <math>\{a, b, c\} \approx \{1, 2, 3\}</math>
|-
| Elements Must Match? || Yes || No
|}

== Conclusion ==

'''Equal sets''' are sets that contain exactly the same elements, regardless of order or repetition. This concept is crucial in understanding set operations, relations, and functions. Mastering it helps students build a strong foundation in mathematics.

[[Category:Set Theory]]
[[Category:Mathematics Class 10]]
[[Category:Mathematics Class 12]]
[[Category:Types of Sets]]