Equal Set vs. Equivalent Set

Attribute	Equal Set	Equivalent Set
Definition	Two sets are equal if they have exactly the same elements.	Two sets are equivalent if they have the same number of elements.
Symbol	=	≡
Example	{1, 2, 3} = {3, 2, 1}	{1, 2, 3} ≡ {a, b, c}
Order	Order of elements does matter.	Order of elements does not matter.

Definition

Equal sets are sets that have exactly the same elements, regardless of the order in which they are listed. For example, {1, 2, 3} is equal to {3, 2, 1}. On the other hand, equivalent sets are sets that have the same number of elements, but the elements themselves may be different. For example, {1, 2, 3} is equivalent to {4, 5, 6} because they both contain three elements.

Cardinality

Equal sets always have the same cardinality, which is the number of elements in the set. If two sets are equal, then they have the same cardinality. For example, if set A = {1, 2, 3} and set B = {3, 2, 1}, then both sets have a cardinality of 3. On the other hand, equivalent sets may or may not have the same cardinality. For example, set C = {1, 2, 3} has a cardinality of 3, while set D = {4, 5, 6, 7} has a cardinality of 4. However, both sets are equivalent because they both contain the same number of elements.

Representation

Equal sets can be represented in different ways, such as using set notation or listing the elements within curly braces. For example, the set {1, 2, 3} can also be written as {3, 2, 1} without changing its meaning. Equivalent sets can also be represented in various ways, but the key difference is that the elements themselves may be different. For example, the sets {1, 2, 3} and {4, 5, 6} are equivalent, even though the elements are not the same.

Operations

When performing operations on sets, it is important to consider whether the sets are equal or equivalent. For equal sets, operations such as union, intersection, and set difference will yield the same result because the elements are exactly the same. For example, if set A = {1, 2, 3} and set B = {3, 2, 1}, then A ∪ B = {1, 2, 3} and A ∩ B = {1, 2, 3}. On the other hand, when dealing with equivalent sets, the results of set operations may vary because the elements themselves are different. For example, if set C = {1, 2, 3} and set D = {4, 5, 6}, then C ∪ D = {1, 2, 3, 4, 5, 6} and C ∩ D = ∅ (empty set).

Examples

Let's consider some examples to further illustrate the difference between equal sets and equivalent sets.

• Equal sets: {1, 2, 3} = {3, 2, 1}
• Equivalent sets: {1, 2, 3} ≡ {4, 5, 6}

In the first example, the sets are equal because they contain the same elements, just in a different order. In the second example, the sets are equivalent because they have the same number of elements, even though the elements themselves are different.

Conclusion

Equal sets and equivalent sets are both important concepts in set theory, but they have distinct attributes that differentiate them. Equal sets have exactly the same elements, while equivalent sets have the same number of elements but the elements themselves may be different. Understanding the difference between these two concepts is crucial when working with sets and performing set operations. By recognizing the characteristics of equal sets and equivalent sets, mathematicians can accurately analyze and manipulate sets in various mathematical contexts.