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Questions and Answers
Which of the following best defines a relation from set X to Y?
What property states that the order of operations does not affect the outcome in binary operations?
Given the relation {(0, -5), (1, -4), (2, -3), (3, -2), (4, -1), (5, 0)}, which statement about this relation is true?
Which operation is represented by (f ο g)(x)?
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What is the domain of the relation R defined by A = {1, 2} and B = {1, 2, 3} with the condition x < y?
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Which of the following properties involves an operation that combines two elements to produce a third element within the same set?
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In the context of functions, which statement correctly defines an inverse?
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If g(x) = 2x + 3 and f(x) = x^2, what is (f · g)(2)?
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What is meant by territoriality in the Philippine context?
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Which of the following are included in the concept of territoriality in the Philippines? (Select all that apply)
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According to the English rule adapted by the Philippines, all cases are triable within Philippine territory except for internal management issues.
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What is the key characteristic of prospectivity in law?
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What is a felony?
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Which type of felony involves deliberate intent?
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What negates the requisite of freedom in felony?
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What is the main distinction between motive and intent?
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In special penal laws, intent is always necessary.
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What is the doctrine of proximate cause?
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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.
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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.
