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.

 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:

$$|A\cup B|=|A|+|B|-|A\cap B|$$

Intersection Formula:

$$|A\cap B|=|A|+|B|-|A\cup B|$$

Complement Formula:

∣Ac∣=U−∣A∣|A^c| = U – |A|∣Ac∣=U−∣A∣

(where U is the universal set)

For Three Sets (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.

$$\max (|A\cap B|)=\min (|A|,|B|)$$

Case 2: Finding Minimum Value of |A ∩ B|
 The minimum value of |A ∩ B| occurs when both sets are disjoint.

$$\min (|A\cap B|)=0$$

Case 3: Finding Maximum Value of |A ∪ B|
 The maximum value of |A ∪ B| occurs when the two sets have no common elements.

$$\max (|A\cup B|)=|A|+|B|$$

Case 4: Finding Minimum Value of |A ∪ B|
 The minimum value of |A ∪ B| occurs when one set is completely inside the other.

$$\min (|A\cup B|)=\max (|A|,|B|)$$

 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,

$$\max (|A\cap B|)=\min (10,7)=7$$

Minimum: When A and B are disjoint,

$$\min (|A\cap B|)=0$$

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.

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