1. What Is a Proper Subset?
A proper subset is a set that contains some elements of another set but not all of them. In other words, set A is a proper subset of set B when:
- Every element of A is present in B
- A and B are not equal
1.1. Notation
The symbol for a proper subset is:
\( A \subset B \)
This means A is fully inside B but smaller than B.
2. Examples of Proper Subsets
These examples show how a proper subset works in simple cases.
2.1. Basic Examples
\( \{1, 2\} \subset \{1, 2, 3\} \)
- \{a, b\} is a proper subset of \{a, b, c\}
- \{blue\} is a proper subset of \{blue, red, green\}
2.2. Examples That Are Not Proper Subsets
- → Because the sets are equal
\( \{1,2,3\} \not\subset \{1,2,3\} \)
- → x is missing in the second set
\{x, y\} \not\subset \{y, z\} \)
3. Difference Between Subset and Proper Subset
A subset can be equal to the set it belongs to, but a proper subset can never be equal to the larger set.
3.1. Comparison Table
| Subset (\( \subseteq \)) | Proper Subset (\( \subset \)) |
|---|---|
| May include all elements of the other set. | Never includes all elements. |
| A can be equal to B. | A must be smaller than B. |
| Example: \( \{1,2,3\} \subseteq \{1,2,3\} \) | Example: \( \{1,2\} \subset \{1,2,3\} \) |
4. How to Check If a Set Is a Proper Subset
To see whether A is a proper subset of B:
4.1. Steps
- Check that all elements of A are present in B.
- Check that A and B are not equal.
- If both conditions are satisfied, A is a proper subset of B.
4.2. Example
\( A = \{2,4\}, B = \{2,4,6,8\} \)
All elements of A are in B, and B has extra elements → A is a proper subset.
5. Important Notes
Some things to keep in mind:
5.1. Key Ideas
- Every proper subset is also a subset, but not every subset is a proper subset.
- A proper subset is always strictly smaller than the set it belongs to.
- The empty set is a proper subset of every non-empty set.