Answered

"5 es divisor de 10 y 9 no es múltiplo de 3" se representa:

$p \vee \sim q$



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AVerdadero
BFalso

Verified Answer :

  • The proposition \(p\) "5 es divisor de 10" is True.
  • The proposition \(q\) "9 es múltiplo de 3" is True, so "\(\sim q\)" (9 no es múltiplo de 3) is False.
  • The word "y" (and) represents the logical conjunction \(\wedge\), not the disjunction \(\vee\).
  • The correct representation is \(p \wedge \sim q\), making the proposed representation \(p \vee \sim q\) incorrect.
  • The final answer is \(\boxed{Falso}\).

Explanation

  1. Analysis of Propositions and Logical Connectors

The statement is composed of two elementary propositions, \(p\) and \(q\).

  • Proposition \(p\): "5 es divisor de 10" (5 is a divisor of 10). Since \(10 \div 5 = 2\), the proposition \(p\) is True.
  • Proposition \(q\): "9 es múltiplo de 3" (9 is a multiple of 3). Since \(3 \times 3 = 9\), the proposition \(q\) is True.

The expression we need to represent is "5 es divisor de 10 y 9 no es múltiplo de 3". This translates to "5 is a divisor of 10" (p) AND (\(\wedge\)) "9 is NOT a multiple of 3" (\(\sim q\)). Thus, the logical representation should be \(p \wedge \sim q\).

  1. Evaluating the Proposed Representation

The proposed representation in the problem is \(p \vee \sim q\). The logical connector '\(\vee\)' represents the disjunction "OR". However, the statement uses the connector "y", which translates to the conjunction "AND" (\(\wedge\)). Therefore, the proposed representation \(p \vee \sim q\) is incorrect because it uses the wrong logical operator. Consequently, the statement that the expression is represented by \(p \vee \sim q\) is false.

Examples

In computer science and digital circuit design, using the correct logical operators is critical. If a system requires two safety sensors to be activated simultaneously to trigger an emergency stop, using an 'OR' operator instead of an 'AND' operator would cause the system to stop if either sensor is triggered, likely leading to unnecessary downtime. This simple translation exercise demonstrates the importance of precision when converting human instructions into logical expressions.