Understanding the ⊂ Symbol

The ⊂ symbol is used to denote that one set is a proper subset of another set. This means that every element of the first set is also an element of the second set, but the two sets are not identical.

The symbol ⊂ (U+2282) specifically refers to a "proper subset." This is a fundamental concept in set theory, a branch of mathematics.

• Definition: As stated in the provided reference, the symbol "\u2282" means "is a proper subset of."
• Usage: If you write A ⊂ B, it means that all elements of set A are also elements of set B, and set B contains at least one element that is not in set A. In simpler terms, A is a part of B, but B is strictly "larger" than A.

Proper Subset (⊂) Explained

When we say a set is a proper subset, it implies a strict inclusion.

Example:
Let's consider two sets:

• Set A = {1, 2, 3}
• Set B = {1, 2, 3, 4, 5}

In this case:

• Every element in set A ({1, 2, 3}) is also present in set B.
• Set B contains elements (4, 5) that are not in set A.
• Therefore, A is a proper subset of B, written as A ⊂ B.

The number of elements in a proper subset is always strictly less than the number of elements in the superset (i.e., n(A) < n(B)).

Distinguishing from ⊆ (Subset)

It's crucial to understand the difference between the ⊂ (proper subset) and ⊆ (subset) symbols, as they are often confused. The provided reference specifically highlights the ⊆ symbol:

• Reference Note: "Since all of the members of set A are members of set D, A is a subset of D. Symbolically this is represented as A ⊆ D. Note that A ⊆ D implies that n(A) ≤ n(D) (i.e. 3 ≤ 6)."

This means:

• A ⊆ D: All members of set A are members of set D. This includes the possibility that A and D could be the exact same set.
• A ⊂ D: All members of set A are members of set D, but A and D cannot be the exact same set. D must contain at least one element not in A.

Comparison Table: ⊂ vs. ⊆

To clarify their uses, here's a direct comparison:

Feature	⊂ (Proper Subset of)	⊆ (Subset of)
Meaning	Set A is strictly contained within Set B.	Set A is contained within or equal to Set B.
Equivalency	A cannot be equal to B (A ≠ B).	A can be equal to B (A = B).
Cardinality	The number of elements in A is strictly less than B: n(A) < n(B).	The number of elements in A is less than or equal to B: n(A) ≤ n(B) (as noted in the reference).
Symbolic Form	A ⊂ B	A ⊆ B
Reference Note	"The symbol "\u2282" means "is a proper subset of"."	"A is a subset of D. Symbolically this is represented as A ⊆ D."

Practical Insights and Examples

• Using ⊂:
  • If Set Fruits = {Apple, Banana} and Set HealthyFoods = {Apple, Banana, Carrot, Orange}.
    • Then Fruits ⊂ HealthyFoods because all fruits are healthy foods, but not all healthy foods are fruits (e.g., Carrot).
• Using ⊆:
  • If Set EvenNumbers = {2, 4, 6} and Set AllNumbers = {1, 2, 3, 4, 5, 6, 7}.
    • Then EvenNumbers ⊆ AllNumbers.
  • If Set MyBooks = {Math, Physics} and Set YourBooks = {Math, Physics}.
    • Then MyBooks ⊆ YourBooks is true because all elements of MyBooks are in YourBooks, and they happen to be the same set. MyBooks ⊂ YourBooks would be false here.

The ⊂ symbol is essential for precisely defining the relationship between sets where one is a true, smaller part of another.