Course: MATH 1013 Math in the Modern World
Week 3: Introduction to Sets
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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.
At the end of the lesson, students should be able to:
Describe sets, subsets, null set, and cardinality of sets.
Translate sets in three notations: descriptive, tabular, and set-builder form.
Differentiate between a subset and a proper subset.
Distinguish between finite and infinite sets.
Determine if two sets are equal or equivalent by examining their elements.
A set is defined as a collection of distinct objects.
Example:
A = {1, ◇, □}
B = {A, O}
A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
B = {M, A, T, H, E, I, C, S} - the letters in the word MATHEMATICS.
Z+ = {1, 2, 3, 4, 5,...} - the set of positive integers.
C = {7, 8, 9, 10, ...} - integers more than 6.
Elements are the objects listed within a set, separated by commas.
Example of elements in B: M, A, T, H, E, I, C, S.
Descriptive or Rule Notation:
Example: A = {Michelle, Mara, Marielle, Mica, Meryl}.
Tabular or Listing Form:
Example: B = {3, 2} for the values satisfying 2−5 + 6 = 0.
Set-builder Form:
Example: C = {x | x is a prime number}.
Denoted as Ø or A = { }.
A set with no elements, also called empty, null, or void set.
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.
A = {a, b, a, b} - Finite
B = {1, 3, 4, 7, 9} - Finite
C = {0, 2, 4, 6, 8, ...} - Infinite (continues indefinitely)
Negative integers - Infinite
Integers between -10 and 10 - Finite
Set A is a subset of B (A ⊆ B) if all elements in A are also elements of B.
A = {1, 2, 3}; B = {1, 2, 3, 4, 5, 6, 7, 8}.
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.
A = {2, 3, 4, 5}; B = {1, 2, 3, 4, 5, 6, 7, 8}.
{2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {}
{2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {}
Two sets A and B are equal (A = B) if they contain the exact same elements.
A = {1, 2, 3}; B = {1, 2, 3, 1, 2, 3, ...} - Same elements.
Sets A and B are equivalent (A ~ B) if they contain the same number of elements.
V = {M, A, L, E}; D = {L, A, M, E} - Same number of elements, thus equivalent.
If set A has n elements, the number of subsets is 2^n.
The lesson provides a foundational understanding of sets, their properties, and classifications crucial for further advanced mathematical studies.
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.