Understanding Symmetry and Mathematics in Life

Symmetry

• Objects can be divided into two identical halves by an imaginary line.

Rotational Symmetry

• Rotated around a central point that appears two or more times.
• Called order and is still identical to its original position.

Point Symmetry

• Every part has a matching part that is equidistant from the central point but in the opposite direction.

Importance of Mathematics in Life

• Restaurant tipping
• Netflix film viewing
• Calculating bills
• Computing test scores
• Doing exercise
• Handling money
• Making countdowns
• Baking and cooking
• Surfing the internet

Appreciating Mathematics As A Human Endeavor

• Agriculturists
• Architects
• Biologists
• Chemists
• Computer programmers
• Engineers (Chemical, Civil, Electrical, Industrial, Material)
• Lawyers
• Managers
• Medical doctors
• Meteorologists
• Politicians
• Salespeople
• Technicians

Fibonacci Sequence

• Series of numbers where each number is the sum of the two preceding numbers.
• Called a Fibonacci number.
• The sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Introduction to Fibonacci Sequence

• F = {1,1,2,3,3,4,4,4} - F = {4}
• {{a}, {a,b}}

Infinite Set

• B = {x∈ Z+}

Power Set (P(n))

• Set of all the subsets of the set.
• Example: A = {1,2,3}
• P(A) = { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
• { } = null set or “Ø” empty set.
• Formula: 2n = where “n” is the sum of all the elements in the sets
• Example: P(A) = 2(3) = 222 = 8

Universal Set (U)

• Usually denoted by the capital letter ‘U’, sometimes by ε(epsilon). A set that contains all the elements of other sets including its own elements.
• Example: U = {x|0 < x < 10}
• U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Intersection (⋂)

• Objects that belong to A and B but not to their intersection.

Symmetric Differences (⊕)

• Objects that belong to A or B but not to their intersection.
• If the two sets are A and B, then symmetric differences belong to both A or B but not all.
• Example 1: A = {a, b, c, d}, B = {a, c, e, f, g}, A⊕B = {b, d, e, f, g}
• Example 2: S = {-7, 12, 76, 315, 426, 900}, T = {34, -7, 89 ,315, 900}, S⊕T = {12, 34 ,76, 89, 426}

Algebraic Properties of Sets

• Commutative Properties:
  • Union and intersection operation satisfy the commutative property, which means that p+q=q+p in algebraic terms.
  • (A⋃B)⋃C = A⋃(B⋃C)
  • (A⋂B)⋂C = A⋂(B⋂C)
• Associative Properties:
  • Union and intersection fulfill the associative property of sets.
  • (A⋃B)⋃C = A⋃(B⋃C)
  • (A⋂B)⋂C = A⋂(B⋂C)
• Distributive Properties:
  • A⋃(B⋂C) = (A⋃B)⋂(A⋃C)
  • A⋂(B⋃C) = (A⋂B)⋃(A⋂C)
• Idempotent:
  • A⋃A = A
  • A⋂A = A
• Complement (De Morgan’s Laws):
  • A⋂A’ = ∅
  • A⋃A’ = U
  • For any two finite sets A and B;
    • (i) A – (B ∩ C) = (A – B) U (A – C)
    • (ii)A – (B U C) = (A – B) ∩ (A – C)
  • De Morgan’s Laws can also be written as:
    • (i)(A ∩ B)’ = A’ U B’
    • (ii) (A U B)’ = A’ ∩ B’

Addition Principle

• How many different ways can something happen, if there are multiple stages involved?
• Example: There are ten students in the class and the teacher wants to pick three students to stand at the front of the class. How many different ways are there to pick the students?
  • In this example, there are three stages: picking the first student, picking the second student, and picking the third student.
  • For each stage, there is a choice of 10 students.
  • Therefore, there are 10 * 10 * 10 = 1000 ways to pick three students.

The Domain and Co-domain of a Relation

• The domain of a relation is the set of all first elements of the ordered pairs in the relation.
• The co-domain of a relation is the set of all second elements of the ordered pairs in the relation.
• Example: Let R be a relation from the set A = {1, 2} to the set B = {1, 2, 3}. Suppose the Relation R is defined as R = {(1, 1), (2, 2)}. Then the domain of R is {1, 2} and the co-domain is {1, 2, 3}.

Functions

• A function is a special kind of relation where each element in the domain is paired with exactly one element in the co-domain.
• Properties of Functions:
  1. Every element of A is the first element of an ordered pair of F.
  2. No two distinct ordered pairs in F have the same first element.
• Example of a function: The relation F = {(1, 1), (2, 2)} is a function because each element in the domain {1, 2} is paired with exactly one element in the co-domain {1, 2, 3}.

Arrow Diagrams of Relations

• An arrow diagram is a way of representing a relation visually.
• In an arrow diagram, the elements of the domain are represented by points on the left side of the diagram, and the elements of the co-domain are represented by points on the right side of the diagram.
• An arrow is drawn from an element in the domain to an element in the co-domain if and only if the two elements are related.
• Example: In the example above, the arrow diagram for the relation R would look like this:
  • left side: 1, 2
  • Right side: 1, 2, 3
  • An arrow would be drawn from 1 to 1 and from 2 to 2.

Binary Operations

• A binary operation is a function that takes two inputs and produces one output.
• Example: The operation of addition is a binary operation because it takes two numbers as input and produces one number as output.

Graph of Functions

• The graph of a function is a set of points in the plane.
• Each point represents a value in the domain of the function paired with its corresponding value in the co-domain.
• Vertical Line Test: A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.

Simple Statement

• Single idea.
• Example: Daniel attends the opera concert.

Compound Statement

• Multiple ideas connected by connectives.
• Example: Daniel attends the opera concert or Lyka performs in the opera concert.

Logic

• Study of reasoning methods.
• Analysis of reasoning methods is Logic’s focus.
• Logic is interested in form, not content.
• Two branches of logic:
  • Mathematical logic - symbolic or formal logic.
  • Informal logic - Another branch of logic.

Proposition

• True or false sentence.
• Another name for proposition is declarative sentence.
• Example of propositions:
  • Gwen performs in the opera concert.
  • Kim attends the opera concert.

Connectives

• Combined two or more propositions.
• Examples:
  • Gwen performs in the opera concert and Kim attends the opera concert.
  • Lyka performs in the opera concert or Gwen performs in the opera concert.

Negation

• The negation of a statement is a statement that is true when the original statement is false, and false when the original statement is true.
• Example: The negation of the statement “Lyka performs in the opera” is “Lyka will not perform in the opera.”

Quantifiers

• Express how many “objects” satisfy a given property or idea.
• Words like “all,” “there exists,” and “none.”
• Quantified Statements:
  • A statement with at least one quantifier.

Variable

• Represents an unspecified object.

This quiz explores various types of symmetry, including reflective, rotational, and point symmetry. It also emphasizes the importance of mathematics in everyday scenarios and various professions, alongside an introduction to the Fibonacci sequence. Test your knowledge of how mathematics intersects with practical life and various fields!