Elige la forma simbólica que represente la siguiente afirmación formada con las proposiciones lógicas p, q, r que se muestran.
[tex]\left.\begin{array}{l}\text { p.: Estálloviendo } \\ \text { q: Iré al cine } \\ \text { r. Tengo tiempo }\end{array}\right] \xrightarrow{\text { Si no está lloviendo y tengo tiempo, entonces iré al cine }}[/tex]
A. [tex]$\neg(r \vee q) \rightarrow q$[/tex]
B. [tex]$(-p \wedge r) \rightarrow q$[/tex]
C. [tex]$(q \rightarrow r) \wedge(r \rightarrow p)$[/tex]
D. [tex]$q \leftrightarrow(r \wedge \neg p)$[/tex]
Answer (2)
Translate 'not p' as ¬ p .
Translate 'not p and r' as ( ¬ p ) ∧ r .
Translate 'If not p and r, then q' as (( ¬ p ) ∧ r ) → q .
The correct answer is ( ¬ p ∧ r ) → q , which corresponds to option B. B
Explanation
Understanding the Statement Let's break down the given logical statement into symbolic form. We have three propositions:
p: It is raining. q: I will go to the cinema. r: I have time.
The statement is: "If it is not raining and I have time, then I will go to the cinema."
We need to translate this into symbolic logic.
Translating to Symbolic Form First, "it is not raining" is the negation of p, which is written as ¬ p .
Next, "it is not raining and I have time" is the conjunction of ¬ p and r, which is written as ( ¬ p ) ∧ r .
Finally, "If it is not raining and I have time, then I will go to the cinema" is a conditional statement, where ( ¬ p ) ∧ r is the antecedent and q is the consequent. This is written as (( ¬ p ) ∧ r ) → q .
Matching with the Options Now, we compare our symbolic form (( ¬ p ) ∧ r ) → q with the given options:
A. ¬ ( r ∨ q ) → q B. ( ¬ p ∧ r ) → q C. ( q → r ) ∧ ( r → p ) D. q ↔ ( r ∧ ¬ p )
Option B matches our symbolic form exactly.
Final Answer Therefore, the correct symbolic representation of the given statement is ( ¬ p ∧ r ) → q .
Examples
Logical statements like this are used in computer programming to control the flow of execution. For example, a program might only execute a certain block of code if a condition is met, such as 'if the user is logged in and has sufficient permissions, then allow access to the sensitive data'. This ensures that the program behaves correctly and securely based on different conditions.
La forma simbólica que representa la afirmación "Si no está lloviendo y tengo tiempo, entonces iré al cine" es ( ¬ p ∧ r ) → q . La opción correcta es la B. B
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