Sets, Relations & Functions
Logic and Statements
Grade 11
Question:
<p>Which of the following is not logically equivalent to the following proposition? 'A real number is either rational or irrational'</p>
<p>(a) If a number is neither rational nor irrational, then it is not real</p>
<p>(b) If a number is not a rational or not an irrational, then it is not real</p>
<p>(c) If a number is not real, then it is neither rational or irrational</p>
Step-by-Step Solution
Key Concept: Logical equivalence requires careful analysis of De Morgan's laws and the structure of implications.
<p>The original statement is 'A real number is either rational or irrational', which means: If a number is real, then it is rational or irrational.</p><p>Option (a) is the contrapositive: If a number is neither rational nor irrational, then it is not real. ✓ Logically equivalent.</p><p>Option (b) states 'If a number is not rational or not irrational', which is logically different from the original statement. ✗ Not equivalent.</p><p>Option (c) is the converse, which may not be equivalent in all cases.</p><p>∴ Answer is (b).</p>
Correct Answer: B