Mathematics in the Modern World: Lesson 5

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which of the following best defines a relation from set X to Y?

A set of ordered pairs where each element in X may relate to multiple elements in Y. (correct)

A set of ordered pairs where one element from Y corresponds to multiple elements in X.

A set of ordered pairs where every element in X is related to one and only one element in Y.

A set of ordered pairs that includes every element of set X and Y.

What property states that the order of operations does not affect the outcome in binary operations?

Associative Property

Closure Property

Distributive Property

Commutative Property (correct)

Given the relation {(0, -5), (1, -4), (2, -3), (3, -2), (4, -1), (5, 0)}, which statement about this relation is true?

It is not a function because one output can relate to multiple inputs.

It is not a function because multiple elements in the first component are the same.

It is a function because each input corresponds to exactly one output. (correct)

It is a function because it includes negative outputs.

Which operation is represented by (f ο g)(x)?

The composition of two functions at x. Signup and view all the answers

What is the domain of the relation R defined by A = {1, 2} and B = {1, 2, 3} with the condition x < y?

{1, 2} Signup and view all the answers

Which of the following properties involves an operation that combines two elements to produce a third element within the same set?

Closure Signup and view all the answers

In the context of functions, which statement correctly defines an inverse?

An inverse function maps every output back to its original input. Signup and view all the answers

If g(x) = 2x + 3 and f(x) = x^2, what is (f · g)(2)?

28 Signup and view all the answers

What is meant by territoriality in the Philippine context?

It refers to laws that are enforceable only within Philippine territory. Signup and view all the answers

Which of the following are included in the concept of territoriality in the Philippines? (Select all that apply)

Public officers performing their official functions Signup and view all the answers

According to the English rule adapted by the Philippines, all cases are triable within Philippine territory except for internal management issues.

True Signup and view all the answers

What is the key characteristic of prospectivity in law?

The law looks forward Signup and view all the answers

What is a felony?

Acts or omissions punishable by law. Signup and view all the answers

Which type of felony involves deliberate intent?

Intentional felony Signup and view all the answers

What negates the requisite of freedom in felony?

Irresistible force or uncontrollable fear. Signup and view all the answers

What is the main distinction between motive and intent?

Intent is the purpose to use particular means to effect a result, while motive is the reason to commit the act. Signup and view all the answers

In special penal laws, intent is always necessary.

False Signup and view all the answers

What is the doctrine of proximate cause?

It states that one is criminally liable for the direct, natural, or logical consequences of their unlawful act. Signup and view all the answers

Study Notes

Relations

A relation from set X to set Y consists of ordered pairs (x, y) where each element x in set X corresponds to at least one element in set Y.
Example sets: A = {1, 2, 3} and B = {2, 3, 4}. Here, x in A is related to y in B if x is less than y.

Functions

A function is a specific type of relation where every input is linked to exactly one output.
A function from set X to Y represents ordered pairs (x, y) without repeating the first component for distinct pairs.

Operations on Functions

Addition and Subtraction: ((f ± g)(x) = f(x) ± g(x))
Multiplication: ((f·g)(x) = f(x)·g(x))
Division: (f(x)/g(x)), where (g(x) \neq 0)
Composition: ((f ο g)(x) = f(g(x)))

Properties of Binary Operations

Closure Property: An operation results in an element within the same set.
Commutative Property: Order of operation does not affect the result.
Associative Property: Grouping of elements does not impact the outcome.
Distributive Property: Operation distributes over the elements.
Identity Elements: An element that, when used in an operation with another element, returns the other element.
Inverses: An element that combines with another to produce the identity element.

Exercises

Exercise 1 involves determining the relationships in sets A = {1, 2} and B = {1, 2, 3}, checking conditions of R.
Exercise 2 focuses on drawing arrow diagrams to represent relations between elements in sets A = {1, 2, 3} and B = {1, 3, 5}.
Exercise 3 requires determining if given relations qualify as functions, notably checking pairs and equations like 2x + 3y – 1 = 0.

Arrow Diagrams

Arrow diagrams visually represent relationships between elements of a set, mapping input to output clearly.

Territoriality of Philippine Law

Enforceable only within Philippine territory, which includes land, airspace, interior waters, and a maritime zone extending 12 miles from the coastline.
Territoriality encompasses Philippine ships, airships, vessels, aircraft registered with the Bureau of Customs.
Forgery of currency, public officers' functions, and crimes against national security fall within territorial enforcement.

Legal Rules and Principles

English Rule: All crimes are triable within Philippine territory, except internal management issues.
French Rule: Crimes affecting external peace and security are also triable.

Prospectivity Characteristics

Laws cannot have retroactive effects; they must look forward.
Favorable laws for the accused may be considered retroactive, with exceptions for habitual delinquents.

Theories of Criminal Responsibility

Classical Theory: Focuses on the act's result, human free will, and retribution.
Positivist Theory: Views crime as a social and moral phenomenon, requiring prior investigation and psychological assessment; emphasizes the enforcement of individual measures.

Revised Penal Code (RPC)

Based on classical theory, includes positivist elements in specific articles (e.g., Article 4, Article 13).
Took effect on January 1, 1932.
Defines felony (delito) as acts or omissions punishable by law and differentiates between deliberate deceit (dolo) and fault (culpa).

Commission and Omission

Commission: The act of violating a law that forbids an action.
Omission: Failure to act when required by law.

Types of Felony

Intentional Felony: Conducting actions with deliberate intent to cause harm.
Culpable Felony: Unintentional acts performed without malice.

Requisites of Felony and Culpa

Freedom: Can be negated by irresistible force or uncontrollable fear.
Intelligence: Can be negated by minority, insanity, or imbecility.
Intent: Can be negated by mistake of fact.

Intent and Motive

Intent is always presumed in unlawful acts; it can be general (presumed) or specific (must be proven).
Motive explains the reason behind committing a specific act.

Crimes Without Criminal Intent

Includes culpable felonies and regulatory laws (mala prohibita).

Imprudence and Negligence

Imprudence: Lack of skill or insufficient consideration; occurs in hasty actions.
Negligence: Deficiency in perception; occurs in complacent actions.

Sudden Emergency Rule

Applies when a situation arises without sufficient time for consideration and is not caused by one's own negligence.

Mistake of Facts

Criminal liability under lawful belief of an act being legal, despite the mistake not being due to fault.

Types of Criminal Liability

Incurred by mistakes in identity, mistakes in actions (aberratio ictus), or actions resulting in greater harm than intended.
Impossibility of an act applies to crimes against person/property.

Doctrine of Proximate Cause

Establishes that one is liable for the natural and direct consequences of their unlawful act.
Liability extends to creating a sense of danger that leads to injury.

Key Quotes on Proximate Cause

"He who causes the cause is the cause of the evil."
Liability arises from direct and logical consequences of one's actions.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers key concepts from Mathematics in the Modern World, focusing on Relations, Functions, and Binary Operations. Dive into ordered pairs and equations such as y^2=x and x^2+y^2=1 to test your understanding of these mathematical ideas.