SUBSET meaning and definition

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What Does Subset Mean?

In mathematics, a subset is a set that contains only some of the elements from another set. In other words, a subset is a collection of objects or elements that are part of a larger set. Understanding subsets is essential in various branches of mathematics, including algebra, geometry, and number theory.

Definition of a Subset

A subset A of a set B is denoted as A ⊆ B or A ≤ B. In this notation, A is the subset and B is the superset (or universal set). To put it simply, every element in A must also be an element in B. For example, if we have a set of numbers {1, 2, 3, 4, 5}, we can create subsets such as:

The set of even numbers: {2, 4} ⊆ {1, 2, 3, 4, 5}
The set of prime numbers: {2, 3, 5} ⊆ {1, 2, 3, 4, 5}

Properties of Subsets

Here are some key properties of subsets:

Subset relation: A ⊆ B implies that every element in A is also an element in B.
Equality: A = B if and only if A ⊆ B and B ⊆ A (i.e., two sets are equal if they contain the same elements).
Union: The union of two subsets A and B, denoted as A ∪ B, is the set that contains all elements from both sets.
Intersection: The intersection of two subsets A and B, denoted as A ∩ B, is the set that contains only the elements common to both sets.

Examples of Subsets

To illustrate the concept of subsets, let's consider some examples:

Set of students: Suppose we have a set of all students in a school {John, Jane, Bob, Sarah}. We can create subsets such as:
- The set of boys: {John, Bob} ⊆ {John, Jane, Bob, Sarah}
- The set of girls: {Jane, Sarah} ⊆ {John, Jane, Bob, Sarah}
Set of numbers: Suppose we have a set of all natural numbers ℕ = {1, 2, 3, ...}. We can create subsets such as:
- The set of even numbers: {..., -4, -2, 0, 2, 4, ...} ⊆ ℕ
- The set of prime numbers: {2, 3, 5, 7, 11, ...} ⊆ ℕ

Conclusion

In conclusion, a subset is a set that contains some or all elements from another set. Understanding subsets is crucial in various areas of mathematics, as it helps us to define and analyze sets more effectively. By recognizing the properties and examples of subsets, we can better comprehend complex mathematical concepts and solve problems with ease.

References

"Set Theory" by Herbert B. Enderton
"Mathematics: A Concise Introduction" by David M. Burton

I hope this article helps you understand what a subset is!

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