Notes on Relations and Functions

Introduction

• Discussion on the language of relations and functions.
• Importance of relations in daily life (e.g., relationships between people, business transactions).

Definitions

Relation

• A rule that relates values from a domain to a range.
• Elements of the domain are inputs that generate outputs.
• A relation is a set of ordered pairs (x, y).
• Example: Relation R = {(1, 2), (2, 4), (3, 6), (4, 8), (5, 10)}
  • Domain: {1, 2, 3, 4, 5}
  • Range: {2, 4, 6, 8, 10}

Set Example

Given Sets

• Set A = {1, 2}
• Set B = {1, 2, 3}

Defining Relation R

• Relation R defined by the condition: ( x - y / 2 ) is an integer.
• Determine ordered pairs in A x B that are in relation R:
  • (1, 1): Yes
  • (1, 2): No
  • (1, 3): Yes
  • (2, 1): No
  • (2, 2): Yes
  • (2, 3): No

Elements in Relation R

• Elements of relation R: {(1, 1), (1, 3), (2, 2)}

Questions and Answers

1. Is 1 related to 3?
   • Yes, because (1, 3) is in relation R.
2. Is 2 related to 3?
   • No, because (2, 3) is not in relation R.
3. Is 2 related to 2?
   • Yes, because (2, 2) is in relation R.
4. Domain and Range of R:
   • Domain: {1, 2}
   • Range: {1, 2, 3}

Functions

• A function is a specific type of relation where each element in the domain is related to only one value in the range.
• Functions can be represented in various forms (tables, ordered pairs, graphs, equations).

Identifying Functions

1. Relation F: {(1, 2), (2, 2), (3, 5), (4, 5)}
   • Yes, it is a function (no repeated x values).
2. Relation G: {(1, 3), (1, 4), (2, 5), (2, 6), (3, 7)}
   • No, it is not a function (repeated x values).
3. Relation H: {(1, 3), (2, 6), (3, 9), ...}
   • Yes, it is a function (no repeated x values).

Mapping Diagrams

• A mapping diagram where each x has a unique y is a function (one-to-one).
• If multiple x values map to the same y (many-to-one), it is still a function.
• If one x maps to multiple y values (one-to-many), it is not a function.

Vertical Line Test

• A graph represents a function if each vertical line intersects it at most once.
• If a vertical line intersects at two or more points, it is not a function.

Evaluating Functions

• Example: Q(x) = x² - 2x + 2, evaluate Q(2).
• Steps: Replace x with 2, simplify.

Additional Function Examples

• Function involving substitutions, such as f(3x - 1) = 2x + 1.
• Replace and simplify as needed.

Conclusion

• Importance of understanding relations and functions.
• Encouragement to like, subscribe, and stay updated for more tutorials.