General Aptitude For Gate CSEIT – Set Theory Maxima-Minima Using Short Trick Method Part 1.
General Aptitude For Gate CSEIT – Set Theory Maxima-Minima Using Short Trick Method Part 1.
Contents [hide]
- 0.1 Set Theory – Maxima & Minima Short Tricks for GATE CSE/IT
- 0.2 1. Basic Formulas of Set Theory
- 0.3 2. Maxima and Minima Tricks in Set Theory
- 0.4 3. Shortcut Example Question
- 0.5 Question:
- 0.6 Solution:
- 0.7 4. Short Tricks to Remember
- 0.8 Conclusion
- 0.9 General Aptitude For Gate CSEIT – Set Theory Maxima-Minima Using Short Trick Method Part 1.
- 0.10 GATE Mechanical Resource
- 1
General Aptitude for GATE CSE/IT- 1.1 Set Theory + Maxima-Minima Using Short Trick – Part 1
- 1.2
Part A: Set Theory – GATE-Oriented Concepts - 1.3 Part B: Maxima & Minima (Short Trick Method)
- 1.4
GATE Tip: - 1.5
Want More? - 1.6 General Aptitude For Gate CSEIT – Set Theory Maxima-Minima Using Short Trick Method Part 1.
- 1.7 QUALITATIVE APTITUDE TRICKS & SHORTCUTS FOR
- 1.8 General-GATE-Aptitude-Book.pdf
- 1.9 Maxima and minima
Set Theory – Maxima & Minima Short Tricks for GATE CSE/IT
Set Theory is an important topic in GATE Computer Science & IT. Questions related to Maxima and Minima of sets often appear in Aptitude & Discrete Mathematics.
1. Basic Formulas of Set Theory
For two sets A and B:
Union Formula:
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| – |A \cap B|
Intersection Formula:
∣A∩B∣=∣A∣+∣B∣−∣A∪B∣|A \cap B| = |A| + |B| – |A \cup B|
Complement Formula:
∣Ac∣=U−∣A∣|A^c| = U – |A|
(where U is the universal set)
For Three Sets (A, B, C):
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣B∩C∣−∣C∩A∣+∣A∩B∩C∣|A \cup B \cup C| = |A| + |B| + |C| – |A \cap B| – |B \cap C| – |C \cap A| + |A \cap B \cap C|
2. Maxima and Minima Tricks in Set Theory
Case 1: Finding Maximum Value of |A ∩ B|
The maximum value of |A ∩ B| occurs when one set is completely inside the other.
max(∣A∩B∣)=min(∣A∣,∣B∣)\max(|A \cap B|) = \min(|A|, |B|)
Case 2: Finding Minimum Value of |A ∩ B|
The minimum value of |A ∩ B| occurs when both sets are disjoint.
min(∣A∩B∣)=0\min(|A \cap B|) = 0
Case 3: Finding Maximum Value of |A ∪ B|
The maximum value of |A ∪ B| occurs when the two sets have no common elements.
max(∣A∪B∣)=∣A∣+∣B∣\max(|A \cup B|) = |A| + |B|
Case 4: Finding Minimum Value of |A ∪ B|
The minimum value of |A ∪ B| occurs when one set is completely inside the other.
min(∣A∪B∣)=max(∣A∣,∣B∣)\min(|A \cup B|) = \max(|A|, |B|)
3. Shortcut Example Question
Question:
If |A| = 10 and |B| = 7, then what is the maximum and minimum value of |A ∩ B|?
Solution:
Maximum: When B is completely inside A,
max(∣A∩B∣)=min(10,7)=7\max(|A \cap B|) = \min(10, 7) = 7
Minimum: When A and B are disjoint,
min(∣A∩B∣)=0\min(|A \cap B|) = 0
Final Answer: Maximum = 7, Minimum = 0
4. Short Tricks to Remember
For Intersection (A ∩ B):
- Max = Smaller Set Size
- Min = 0 (if disjoint)
For Union (A ∪ B):
- Max = Sum of Both Set Sizes
- Min = Larger Set Size
Conclusion
These short tricks help solve Set Theory Maxima-Minima problems quickly in GATE, SSC, and other competitive exams. Would you like me to add more practice questions with solutions?
General Aptitude For Gate CSEIT – Set Theory Maxima-Minima Using Short Trick Method Part 1.
GATE Mechanical Resource
Here’s a structured guide for a video lecture or study note on:
General Aptitude for GATE CSE/IT
Set Theory + Maxima-Minima Using Short Trick – Part 1
Part A: Set Theory – GATE-Oriented Concepts
Key Concepts:
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Set Notation: A, B, C are sets. Universal Set: U
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Operations:
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Union: A ∪ B
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Intersection: A ∩ B
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Complement: A′ or U − A
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Difference: A − B
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Venn Diagram Trick (Shortcuts):
Problem: In a class of 100 students, 60 like math, 45 like physics, and 20 like both. Find how many like only math, only physics, and neither.
Trick Formula:
→ Only A = A − (A ∩ B)
→ Only B = B − (A ∩ B)
→ Total = Only A + Only B + Both + Neither
Solution:
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Only Math = 60 − 20 = 40
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Only Physics = 45 − 20 = 25
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Total liking either or both = 40 + 25 + 20 = 85
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Neither = 100 − 85 = 15
Part B: Maxima & Minima (Short Trick Method)
What to Know:
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Maxima/Minima come from differentiation:
Iff′(x) = 0, and:-
f′′(x) > 0⇒ local minimum -
f′′(x) < 0⇒ local maximum
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Short Trick Example:
Problem: What is the maximum value of the function f(x) = −x² + 4x + 1?
Shortcut Steps:
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It’s a quadratic of form
ax² + bx + c -
If a < 0, parabola opens downward ⇒ max at vertex
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Max value at
x = −b/2a = −4/(2×−1) = 2 -
f(2) = −(2)² + 4×2 + 1 = −4 + 8 + 1 = 5
So, Maximum = 5 at x = 2
GATE Tip:
-
For most Set Theory problems, Venn diagram + formula trick works best.
-
For Maxima/Minima in aptitude, always check if it’s a quadratic → use vertex formula
−b/2a
Want More?
Would you like:
-
PDF notes with 5+ solved tricks per topic? -
Lecture slides with animation for GATE-style Venn questions? -
GATE PYQs (Previous Year Questions) from Set Theory & Max-Min?
Let me know, and I’ll prepare a Part 2 with word problems + optimization tricks for aptitude!