Algebra 2 Lesson on Sets
Introduction to Sets
- Sets: A collection of things, typically numbers, with a clear definition of membership.
- Importance: Sets are foundational for topics in algebra and appear in standardized tests like SAT, ACT, and in college-level math.
Definition and Examples
- A set is a collection where membership is clearly defined.
- Example: Whole Numbers
- Defined as numbers starting from 0, increasing by 1 indefinitely.
- Written as {0, 1, 2, 3, ...} with dots indicating continuation.
- Members/Elements: Listed numbers in the set.
Set Notation
- Sets are enclosed in set braces: {}.
- Roster Method: Listing elements within set braces.
Types of Sets
- Integers: A set using roster method: {..., -3, -2, -1, 0, 1, 2, 3, ...}.
- Non-numeric sets: E.g., first five letters of the alphabet: {a, b, c, d, e}.
Properties of Sets
- Order of elements is not important.
- Sets with identical elements are equal regardless of order.
- Example: States bordering Louisiana: Mississippi, Arkansas, Texas.
Naming and Empty Sets
- Sets often named with capital letters, e.g., A, B.
- Empty Set/Null Set: A set with no elements, denoted by ∅ or {}.
- Solution Set Notation: Used for solutions of equations.
- Example: Solution of 2x - 5 = 27 is x = 16, written as {16}.
- No solution case represented by empty set symbols, but do not mix symbols: ∅ ≠{∅}.
Universal Set
- Universal Set (denoted by U): Contains all elements under consideration.
- Example: For states bordering Louisiana, U would be all U.S. states.
Finite and Infinite Sets
- Finite Set: Has a specific number of elements.
- Infinite Set: Elements continue indefinitely.
- Example: Whole numbers larger than 5 is infinite.
Subsets
- Subset: A set where all elements are contained within another set.
- Denoted by ⊆ for subsets, ⊈ for not subsets.
- Proper Subset: All elements of one set in another, but not equal.
- Improper Subset: Two sets are equal.
Practice with Subsets
- Calculate number of subsets with formula
2^n where n is number of elements.
- Example: Set {5, 7, 9} has 8 subsets including the empty set.
This summary provides a comprehensive introduction to the concept of sets, crucial for understanding algebraic foundations.