Sets — Class 11 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Sets" — 8 important questions with detailed answers for CBSE board exam preparation.
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TL;DR: Free step-by-step NCERT solutions for Class 11 Mathematics chapter "Sets" — 8 important questions with detailed answers for CBSE board exam preparatio…
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Key Questions Covered:
- If A = {x : x is a natural number and x < 6} and B = {x : x is an even natura…
- For sets A, B, and C, verify De Morgan's Law: (A ∪ B)' = A' ∩ B' when U = {1,…
- Find the number of subsets and proper subsets of set S = {a, b, c, d}.
- Verify A ⊆ B and find A × B when A = {1, 2}, B = {1, 2, 3, 4}.
- If A = {x : x² - 5x + 6 = 0} and B = {x : x² - 3x + 2 = 0}, find A ∪ B and ve…
- In a class of 50 students, 30 like Mathematics and 25 like Science. If 10 stu…
- + 2 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| If A = {x : x is a natural number and x < 6} and B = {x :… | ✓ Solved |
| For sets A, B, and C, verify De Morgan's Law: (A ∪ B)' = … | ✓ Solved |
| Find the number of subsets and proper subsets of set S = … | ✓ Solved |
| Verify A ⊆ B and find A × B when A = {1, 2}, B = {1, 2, 3… | ✓ Solved |
| If A = {x : x² - 5x + 6 = 0} and B = {x : x² - 3x + 2 = 0… | ✓ Solved |
| In a class of 50 students, 30 like Mathematics and 25 lik… | ✓ Solved |
Showing 6 of 8 questions
Q1: If A = {x : x is a natural number and x < 6} and B = {x : x is an even natural number and x ≤ 8}, find A ∪ B and A ∩ B.
Step 1: Write set A in roster form.
A = {1, 2, 3, 4, 5}
Step 2: Write set B in roster form.
B = {2, 4, 6, 8}
Step 3: Find A ∪ B (all elements in either A or B).
A ∪ B = {1, 2, 3, 4, 5, 6, 8}
Step 4: Find A ∩ B (elements common to both A and B).
A ∩ B = {2, 4}
Final Answer: A ∪ B = {1, 2, 3, 4, 5, 6, 8}, A ∩ B = {2, 4}
Q2: For sets A, B, and C, verify De Morgan's Law: (A ∪ B)' = A' ∩ B' when U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, B = {3, 4, 5}.
Step 1: Find A' and B' with respect to U.
A' = U - A = {4, 5, 6}
B' = U - B = {1, 2, 6}
Step 2: Find A ∪ B.
A ∪ B = {1, 2, 3, 4, 5}
Step 3: Find (A ∪ B)' = U - (A ∪ B).
(A ∪ B)' = {6}
Step 4: Find A' ∩ B'.
A' ∩ B' = {4, 5, 6} ∩ {1, 2, 6} = {6}
Step 5: Verify (A ∪ B)' = A' ∩ B'.
{6} = {6} ✓
Final Answer: De Morgan's Law (A ∪ B)' = A' ∩ B' is verified.
Q3: Find the number of subsets and proper subsets of set S = {a, b, c, d}.
Step 1: Count the elements in set S.
Number of elements n = 4
Step 2: Find total number of subsets.
Total subsets = 2ⁿ = 2⁴ = 16
Step 3: List some subsets to verify.
∅, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, {a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d}
Step 4: Find number of proper subsets.
Proper subsets = Total subsets - 1 (excluding S itself) = 16 - 1 = 15
Final Answer: Total subsets = 16, Proper subsets = 15
Q4: Verify A ⊆ B and find A × B when A = {1, 2}, B = {1, 2, 3, 4}.
Step 1: Check if A ⊆ B.
Every element of A must be in B.
1 ∈ B? Yes
2 ∈ B? Yes
Therefore, A ⊆ B ✓
Step 2: Find A × B (Cartesian product).
A × B = {(a, b) : a ∈ A, b ∈ B}
Step 3: List all ordered pairs.
(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)
Step 4: Write A × B in set notation.
A × B = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4)}
Step 5: Note that |A × B| = |A| × |B| = 2 × 4 = 8
Final Answer: A ⊆ B verified. A × B has 8 ordered pairs as listed above.
Q5: If A = {x : x² - 5x + 6 = 0} and B = {x : x² - 3x + 2 = 0}, find A ∪ B and verify if A ⊆ B.
Step 1: Solve x² - 5x + 6 = 0 to find set A.
(x - 2)(x - 3) = 0
x = 2 or x = 3
A = {2, 3}
Step 2: Solve x² - 3x + 2 = 0 to find set B.
(x - 1)(x - 2) = 0
x = 1 or x = 2
B = {1, 2}
Step 3: Find A ∪ B.
A ∪ B = {1, 2, 3}
Step 4: Verify if A ⊆ B.
Is 2 ∈ B? Yes
Is 3 ∈ B? No
Therefore, A ⊄ B (A is not a subset of B)
Final Answer: A ∪ B = {1, 2, 3}, and A ⊄ B
Q6: In a class of 50 students, 30 like Mathematics and 25 like Science. If 10 students like both, how many students like at least one subject?
Step 1: Define sets.
Let M = set of students who like Mathematics, |M| = 30
Let S = set of students who like Science, |S| = 25
|M ∩ S| = 10 (students liking both)
Step 2: Apply the principle of inclusion-exclusion.
|M ∪ S| = |M| + |S| - |M ∩ S|
|M ∪ S| = 30 + 25 - 10 = 45
Step 3: Interpret the result.
45 students like at least one subject.
Step 4: Find students liking neither.
Students liking neither = Total - |M ∪ S| = 50 - 45 = 5
Final Answer: 45 students like at least one subject; 5 stude...
Showing 6 of 8 questions. Visit the full page for complete solutions.
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