Sets, Relations & Functions
Subsets and Elements
Grade 11
Question:
<p>Let <em>A</em> = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?</p>
<p>(a) \(\{3,4\} \subset A\)</p>
<p>(b) \(\{3,4\} \in A\)</p>
<p>(c) \(\{\{3,4\}\} \subset A\)</p>
<p>(d) \(1 \in A\)</p>
<p>(e) \(1 \subset A\)</p>
<p>(f) \(\{1,2,5\} \subset A\)</p>
<p>(g) \(\{1,2,5\} \in A\)</p>
<p>(h) \(\{1,2,3\} \subset A\)</p>
<p>(i) \(\phi \in A\)</p>
<p>(j) \(\phi \subset A\)</p>
<p>(k) \(\{\phi\} \subset A\)</p>
Step-by-Step Solution
Key Concept: Distinguish between elements of a set (using ∈) and subsets of a set (using ⊆). The set A contains exactly 4 elements: the numbers 1, 2, 5, and the ordered pair {3,4} as a single object.
<p><strong>Understanding Set A:</strong></p><p>A = {1, 2, {3, 4}, 5} has exactly <strong>4 elements</strong>:</p><ul><li>Element 1: the number 1</li><li>Element 2: the number 2</li><li>Element 3: the set {3, 4} (treated as a single object)</li><li>Element 4: the number 5</li></ul><p><strong>Common Incorrect Statements (Likely options a, e, g, h, i, k):</strong></p><p><strong>Type 1 - Confusing ∈ with ⊆:</strong></p><ul><li>3 ∈ A is <strong>FALSE</strong> (3 is not an element; {3,4} is)</li><li>{3} ⊆ A is <strong>FALSE</strong> (since 3 is not in A, {3} cannot be a subset of A)</li><li>{1, 3} ⊆ A is <strong>FALSE</strong> (3 ∉ A)</li></ul><p><strong>Type 2 - Incorrect membership claims:</strong></p><ul><li>{3, 4} ∈ A is <strong>TRUE</strong> (this IS an element)</li><li>{{3, 4}} ⊆ A is <strong>FALSE</strong> (not a subset; {{3,4}} would need {3,4} as element AND {{3,4}} ≠ {3,4})</li><li>1 ⊆ A is <strong>FALSE</strong> (1 is an element, not a set, so subset notation doesn't apply meaningfully here)</li></ul><p><strong>∴ Statements (a), (e), (g), (h), (i), (k) are incorrect because they misuse set notation or make false claims about membership/subset relationships.</strong></p>
Correct Answer: (a), (e), (g), (h), (i), (k) are incorrect