Sets, Relations & Functions
Subsets and complements
Grade 11
Question:
<p>Let <em>A</em> and <em>B</em> be two non-empty subsets of a set <em>X</em> such that <em>A</em> is not a subset of <em>B</em>. Then</p>
<p><em>A</em> is a subset of complement of <em>B</em></p>
<p><em>B</em> is a subset of <em>A</em></p>
<p><em>A</em> and <em>B</em> are disjoint sets</p>
<p><em>A</em> and complement of <em>B</em> are non-disjoint sets</p>
Step-by-Step Solution
Key Concept: The negation of 'A ⊆ B' means there exists at least one element in A that is not in B, i.e., A ⊈ B ⟺ (∃x ∈ A such that x ∉ B). This is equivalent to A ∩ B' ≠ ∅ (where B' is complement of B in X).
<p><strong>Step 1:</strong> Understand the statement 'A is not a subset of B' (written A ⊈ B).</p><p><strong>Step 2:</strong> By definition, A ⊆ B means every element of A is in B. The negation is: A ⊈ B ⟺ ∃x ∈ A such that x ∉ B.</p><p><strong>Step 3:</strong> This means there exists at least one element in A that is not in B, so A ∩ B' ≠ ∅ (or equivalently, A - B ≠ ∅).</p><p><strong>Step 4:</strong> Note that A ∩ B may still be non-empty; we only require the 'excess' part of A (outside B) to be non-empty.</p><p>∴ Answer: D (The correct option states A ∩ B' ≠ ∅ or A - B ≠ ∅, depending on the given choices)</p>
Correct Answer: D