1. What Is a Subset?
A subset is a set whose every element also belongs to another set. If all elements of set A are found in set B, then A is a subset of B.
This idea helps compare sets and understand how they are related.
1.1. Notation
- \( A \subseteq B \) : A is a subset of B
- \( A \subset B \) : A is a proper subset of B
2. Understanding Subsets with Examples
Here are some simple examples to understand how subsets work.
2.1. Basic Examples
\( A = \{1,2\}, B = \{1,2,3,4\} \Rightarrow A \subseteq B \)
- \{a, b\} is a subset of \{a, b, c, d\}
- \{red\} is a subset of \{red, blue, green\}
2.2. Examples Where It Is Not a Subset
- → 5 is not in the second set
\( \{1,5\} \nsubseteq \{1,2,3,4\} \)
- \{x, y\} is not a subset of \{y, z\} because x is missing
3. Proper Subsets
A proper subset contains some elements of the larger set but not all elements. In simple words, A is a proper subset of B when:
- Every element of A is in B
- A and B are not equal
3.1. Examples
\( \{1,2\} \subset \{1,2,3\} \)
- \{a, b\} is a proper subset of \{a, b, c\}
4. Empty Set as a Subset
The empty set is a subset of every set. This is because it has no elements to violate the condition.
4.1. Reason
Since there is nothing inside \( \emptyset \), the statement “every element of \( \emptyset \) is also in A” is always true.
5. How to Check if One Set Is a Subset of Another
To check whether A is a subset of B:
5.1. Steps
- Look at each element of A.
- Check if it is present in B.
- If all elements of A appear in B, then A is a subset of B.
5.2. Example
Check if \( \{2,4\} \) is a subset of \( \{2,4,6,8\} \).
Both 2 and 4 appear in the second set → So it is a subset.
6. Important Points
Some useful things to remember:
6.1. Key Ideas
- Every set is a subset of itself.
- The empty set is a subset of every set.
- If A is a subset of B and B is a subset of C, then A is a subset of C.
- A proper subset always has fewer elements than the set it belongs to.