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:

$A = B ⟺ (x \in A \Rightarrow x \in B) and (x \in B \Rightarrow x \in A)$

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 $A = B$ , then $B = A$ (symmetry).

Symbolic Representation

If $A$ and $B$ are equal sets, we write: $A = B$

Examples of Equal Sets

Example 1:

$A = {1, 2, 3}$ $B = {3, 2, 1}$ A = B because they have the same elements, even though the order is different.

Example 2:

$C = {a, b, c}$ $D = {a, a, b, c}$ C = D because repetition does not change the set. D still contains only a, b, and c.

Example 3:

Let Failed to parse (syntax error): {\displaystyle E = \{\text{vowels in "APPLE"}\} = \{A, P, L, E\}} $F = {A, E, L, P}$ Then E = F because they contain the same distinct letters.

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:

   $A \subseteq B$  and  $B \subseteq A$   
  If both are true, then  $A = B$ .

Difference Between Equal and Equivalent Sets

Property	Equal Sets	Equivalent Sets
Definition	Same elements	Same number of elements
Symbol	$A = B$	$A \approx B$
Example	${1, 2, 3} = {3, 2, 1}$	${a, b, c} \approx {1, 2, 3}$
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.