Two sets are considered equal if and only if they contain exactly the same elements. This concept can be broken down into a formal rule based on set inclusion.
Formal Definition of Set Equality
The core rule for set equality is that two sets, traditionally named A and B, are equal if and only if:
- A is a subset of B (denoted as A ⊆ B). This means every element in set A must also be present in set B.
- B is a subset of A (denoted as B ⊆ A). This means every element in set B must also be present in set A.
If both of these conditions are met, it is stated that A = B. If even one of these conditions is not met, then A ≠ B.
Understanding Set Equality with Examples
Here is a practical breakdown of the rules:
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Identical Elements: Sets must contain exactly the same elements for them to be equal, irrespective of the order in which these elements are presented. For instance, {1, 2, 3} = {3, 2, 1} because they both hold the same three values.
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No Duplicates: Sets do not consider duplicate entries. {1, 1, 2, 3} is not different from {1, 2, 3} and, in mathematical notation, {1, 1, 2, 3} will be simplified to {1, 2, 3}.
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Subset Relationship: The formal definition via subset relations highlights the importance of all elements of each set also being elements of the other. If sets overlap, but some elements are not mutually present, then the sets are not equal.
Summary Table of Set Equality Rules
| Condition | Description | Result |
|---|---|---|
| A ⊆ B | Every element in set A must also be an element in set B. | Necessary but not sufficient on its own. |
| B ⊆ A | Every element in set B must also be an element in set A. | Necessary but not sufficient on its own. |
| A ⊆ B and B ⊆ A | Both sets contain exactly the same elements, and these are mutually contained within the other. | A = B |
| A is not a subset of B, or B is not a subset of A | This implies that the sets do not contain the same elements, that is, there is at least one element that exists in only one set and not in the other | A ≠ B |
Practical Insight: When determining if two sets are equal, check first if the two sets seem similar, and then systematically ensure that each element of the first set exists in the second, and then do the same for the other direction.
Reference: The definition provided in uomustansiriyah.edu.iq states "Two sets, 𝐴 and 𝐵, are said to be equal if and only if A and B contain exactly the same elements and denote that by 𝐴 = 𝐵. That is, 𝐴 = 𝐵 if and only if 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴".
