Sets

Summary

This lesson introduces the concept of sets, focusing on definitions and basic operations. A set is described as a collection of elements, and terms like 'finite sets,' 'infinite sets,' and the 'empty set' are explained. Common number sets such as natural numbers, integers, and rational numbers are discussed. The lesson also covers subsets, proper subsets, unions, and intersections of sets, using examples to clarify these operations. The union includes elements from both sets, while the intersection includes only common elements. Disjoint sets, having no common elements, are also explained.

Takeaways

• 🔢 A *set* is defined as a collection of numbers or objects, and its elements are called *members*.
• ✍️ If X is an element of set P, it is represented using the notation *X ∈ P*.
• ⚪️ An *empty set* has no elements, represented by either { } or a zero with a line through it (∅).
• 🔄 A *finite set* has a countable number of elements, while an *infinite set* has uncountable elements.
• 🔣 Some common sets of numbers include: *Natural Numbers* (0, 1, 2, 3...), *Integers* (positive and negative whole numbers), *Rational Numbers* (P/Q, where Q ≠ 0), and *Real Numbers* (numbers on a real number line).
• 🔍 A *subset* is a set containing some or all elements of another set, without any new elements.
• ☑️ A *proper subset* is a subset that does not contain all elements of the original set.
• 🔗 The *union* of two sets (P ∪ Q) is a set of elements that are in either P, Q, or both, without repetition.
• 🔄 The *intersection* of two sets (P ∩ Q) is the set of elements that are in both P and Q.
• 🚫 If two sets have no elements in common, they are termed *disjoint* or *mutually exclusive*.

Q & A

What is a set in mathematics?

A set is a collection of numbers or objects. The individual items in the set are called elements or members.

How do we represent that an element belongs to a set?

If an element 'X' belongs to a set 'P,' we write 'X' is an element of the set 'P,' using a special symbol to denote membership.

What is an empty set, and how is it notated?

An empty set is a set that contains no elements. It is notated by either brace brackets with nothing inside {} or by the symbol Ø.

What is the difference between a finite set and an infinite set?

A finite set contains a countable number of elements, while an infinite set contains infinitely many elements, making it impossible to count.

What are some examples of commonly used sets of numbers?

Common sets include natural numbers (0, 1, 2, 3, ...), integers (positive and negative whole numbers), rational numbers (P/Q where P and Q are integers and Q ≠ 0), and real numbers (any number that can be plotted on a real number line).

What is a subset?

A subset is a smaller set that contains some or all of the elements of a larger set but no elements outside the original set.

How is a proper subset different from a regular subset?

A proper subset contains some, but not all, of the elements from the original set and no new elements. It is always smaller than the original set.

What is the union of two sets?

The union of two sets P and Q, denoted as P ∪ Q, is the set of all elements that are in P, Q, or both. Each element is listed only once.

What is the intersection of two sets?

The intersection of two sets P and Q, denoted as P ∩ Q, is the set of elements that are common to both P and Q.

What does it mean for two sets to be disjoint or mutually exclusive?

Two sets are disjoint or mutually exclusive if they have no elements in common. Their intersection is the empty set.