Sets, Relations & Functions
Mathematical Reasoning / Negation of Implication
Grade None
Question:
<p>The negation of the statement "If I become a teacher, then I will open a school" is</p>
<p>I will become a teacher and I will not open a school.</p>
<p>Either I will not become a teacher or I will not open a school.</p>
<p>Neither I will become a teacher nor I will open a school.</p>
<p>I will not become a teacher or I will open a school.</p>
Step-by-Step Solution
Key Concept: The negation of a conditional statement (p → q) is NOT (¬p ∨ q), which equals (p ∧ ¬q). The negation requires the hypothesis to be true AND the conclusion to be false simultaneously.
<p><strong>Step 1:</strong> Identify the logical form. Let p = 'I become a teacher' and q = 'I will open a school'. The statement is p → q.</p><p><strong>Step 2:</strong> Recall that ¬(p → q) ≡ p ∧ ¬q. This is because p → q ≡ ¬p ∨ q, so its negation is ¬(¬p ∨ q) = p ∧ ¬q.</p><p><strong>Step 3:</strong> Apply the rule: The negation is 'I become a teacher AND I will not open a school' or equivalently 'I become a teacher but I will not open a school'.</p><p><strong>Step 4:</strong> The correct negation statement is: <em>'I become a teacher and I do not open a school'</em></p><p>∴ Answer: A</p>
Correct Answer: A