Understanding Subsets in Set Theory

What is a Subset?

A subset is a set formed from the elements of another set. If all elements of set A are also elements of set B, then A is considered a subset of B, denoted as A ⊆ B. If A is a subset of B but A is not equal to B, it is called a proper subset, denoted as A ⊂ B.

Types of Subsets

• Proper Subset: A subset that does not contain all the elements of another set. For example, if A = {1, 2}, then A is a proper subset of B = {1, 2, 3}.
• Improper Subset: A subset that is equal to the original set. For example, A = B = {1, 2, 3} is an improper subset.
• Empty Set: The empty set, denoted by ∅, is considered a subset of every set. It contains no elements but is still a valid subset.

Properties of Subsets

• If A ⊆ B, then every element of A is also an element of B.
• The empty set ∅ is a subset of every set.
• Every set is a subset of itself (A ⊆ A).
• If A ⊆ B and B ⊆ A, then A = B.
• If A ⊆ B and B ⊆ C, then A ⊆ C (Transitive property).

Examples of Subsets

To illustrate subsets, let's consider the following example:

Let B = {1, 2, 3, 4}. The possible subsets of B include:

• ∅ (the empty set)
• {1}
• {2}
• {3}
• {4}
• {1, 2}
• {1, 3}
• {1, 4}
• {2, 3}
• {2, 4}
• {3, 4}
• {1, 2, 3}
• {1, 2, 4}
• {1, 3, 4}
• {2, 3, 4}
• {1, 2, 3, 4} (B itself)

In total, there are 2^n subsets for a set with n elements, so set B has 2^4 = 16 different subsets.