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.

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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.