Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Consider the following statements:<br>\(p\) : It rains today<br>\(q\) : I go to school<br>\(r\) : I shall meet any friends<br>\(s\) : I shall go for a movie<br>Then which of the following propositions represents 'If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.'?</p>
<p>\(\sim(p \wedge q) \rightarrow (r \wedge s)\)</p>
<p>\(\sim(p \wedge \sim q) \rightarrow (r \wedge s)\)</p>
<p>\(\sim(p \wedge q) \rightarrow (r \vee s)\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Translate the English statement into logical symbols: 'If A or B, then C and D' becomes (¬p ∨ ¬q) → (r ∧ s). The key is identifying the condition (antecedent) as a disjunction of negations and the consequence (consequent) as a conjunction.
<p><strong>Step 1:</strong> Identify the main structure. The sentence is an 'if-then' statement, so we need a conditional (→).</p><p><strong>Step 2:</strong> Identify the condition (antecedent): 'If it does not rain or if I do not go to school' = (¬p ∨ ¬q)</p><p><strong>Step 3:</strong> Identify the consequence (consequent): 'then I shall meet my friend and go for a movie' = (r ∧ s)</p><p><strong>Step 4:</strong> Combine using the conditional operator: (¬p ∨ ¬q) → (r ∧ s)</p><p>∴ Answer: <strong>(¬p ∨ ¬q) → (r ∧ s)</strong> (or equivalent such as (~p ∨ ~q) → (r ∧ s))</p>
Correct Answer: A