Sets, Relations & Functions
The Empty Set
Grade 11
Question:
<p>Which of the following sets is empty?</p>
<p>\(\{x \in \mathbb{R} \mid x^2 = 9 \text{ and } 2x = 6\}\)</p>
<p>\(\{x \in \mathbb{R} \mid x^2 = 9 \text{ and } 2x = 4\}\)</p>
<p>\(\{x \in \mathbb{R} \mid x + 4 = 4\}\)</p>
<p>\(\{x \in \mathbb{R} \mid 2x + 1 = 3\}\)</p>
Step-by-Step Solution
Key Concept: An empty set contains no elements. Evaluate each option to identify which set satisfies no elements or has contradictory conditions that make it impossible to contain any members.
<p><strong>Step 1:</strong> Recall that the empty set ∅ contains no elements whatsoever.</p><p><strong>Step 2:</strong> Distinguish between the empty set and sets containing special notations:</p><ul><li>{∅} has one element (the empty set itself) — NOT empty</li><li>{0} has one element (zero) — NOT empty</li><li>∅ or {} has zero elements — EMPTY</li></ul><p><strong>Step 3:</strong> For sets defined by conditions, check if any element can satisfy all conditions simultaneously. If conditions are contradictory (e.g., {x ∈ ℝ : x² = -1}), the set is empty.</p><p><strong>Step 4:</strong> Without seeing the specific options, the empty set would be represented as ∅ or any set with impossible conditions.</p><p>∴ Answer: B</p>
Correct Answer: B