LMS-Week-3.0-MATH-1013-Introduction-to-Sets

Introduction to Sets

  • Course: MATH 1013 Math in the Modern World

  • Week 3: Introduction to Sets

  • Copyright Notice: Unauthorized reproduction or sharing of materials is prohibited.

Historical Context

  • The concept of set originated in the 19th century.

  • Developed by George Cantor, a mathematician known for his work on set theory.

  • The German term for set is MENGE.

Learning Outcomes

At the end of the lesson, students should be able to:

  1. Describe sets, subsets, null set, and cardinality of sets.

  2. Translate sets in three notations: descriptive, tabular, and set-builder form.

  3. Differentiate between a subset and a proper subset.

  4. Distinguish between finite and infinite sets.

  5. Determine if two sets are equal or equivalent by examining their elements.

Definition of a Set

  • A set is defined as a collection of distinct objects.

  • Example:

    • A = {1, ◇, □}

    • B = {A, O}

Examples of Sets

Example 1: Days of the Week

  • A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

Example 2: Letters of the Word

  • B = {M, A, T, H, E, I, C, S} - the letters in the word MATHEMATICS.

Example 3: Positive Integers

  • Z+ = {1, 2, 3, 4, 5,...} - the set of positive integers.

Example 4: Integers Greater than 6

  • C = {7, 8, 9, 10, ...} - integers more than 6.

Elements of a Set

  • Elements are the objects listed within a set, separated by commas.

  • Example of elements in B: M, A, T, H, E, I, C, S.

Types of Notation for Sets

Three Different Set Notations:

  1. Descriptive or Rule Notation:

    • Example: A = {Michelle, Mara, Marielle, Mica, Meryl}.

  2. Tabular or Listing Form:

    • Example: B = {3, 2} for the values satisfying 2−5 + 6 = 0.

  3. Set-builder Form:

    • Example: C = {x | x is a prime number}.

Special Types of Sets

Null Set

  • Denoted as Ø or A = { }.

  • A set with no elements, also called empty, null, or void set.

Finite vs Infinite Sets

  • A set is finite if it has a countable number of elements; otherwise, it is infinite.

  • Cardinal number (n(A)) of a finite set indicates the number of elements in set A.

Examples of Finite and Infinite Sets

  1. A = {a, b, a, b} - Finite

  2. B = {1, 3, 4, 7, 9} - Finite

  3. C = {0, 2, 4, 6, 8, ...} - Infinite (continues indefinitely)

  4. Negative integers - Infinite

  5. Integers between -10 and 10 - Finite

Subsets

  • Set A is a subset of B (A ⊆ B) if all elements in A are also elements of B.

Example:

  • A = {1, 2, 3}; B = {1, 2, 3, 4, 5, 6, 7, 8}.

Proper Subsets

  • Set A is a proper subset of B (A ⊂ B) if all elements of A are in B but A is not equal to B.

Example:

  • A = {2, 3, 4, 5}; B = {1, 2, 3, 4, 5, 6, 7, 8}.

Lists of Subsets and Proper Subsets

Subsets of A = {2, 4, 6}

  • {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {}

Proper Subsets of A = {2, 4, 6}

  • {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {}

Equal Sets

  • Two sets A and B are equal (A = B) if they contain the exact same elements.

Example:

  • A = {1, 2, 3}; B = {1, 2, 3, 1, 2, 3, ...} - Same elements.

Equivalent Sets

  • Sets A and B are equivalent (A ~ B) if they contain the same number of elements.

Example:

  • V = {M, A, L, E}; D = {L, A, M, E} - Same number of elements, thus equivalent.

Number of Subsets

  • If set A has n elements, the number of subsets is 2^n.

Example:** SUV Manufacturing Company - 6 Upgrade Options

Conclusion

  • The lesson provides a foundational understanding of sets, their properties, and classifications crucial for further advanced mathematical studies.

References

  • Quintos, R. et al. (2019). Mathematics in the Modern World.

  • Aufmann, R.N. et al. (2018). Mathematics in the Modern World.

  • Additional mathematical resources and texts.

robot