Set Theory Concepts

1. Fundamental Formula Applications

• Formula: For two sets:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

Number in neither:

$$n(\text{Neither})=n(U)-n(A\cup B)$$

Use when two categories overlap and the total population is given.

• Example: In a survey of 150 students, 90 like Math, 70 like Science, and 40 like both. How many like neither? Solution:

        Math          Science
      _________      _________
     /         \____/         \
    /    50      40     30     \
    \___________/  \___________/

Students liking at least one:

$$n(M\cup S)=90+70-40=120$$

Neither:

$$150-120=\boxed{30}$$

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2. Only One Set

• Formula:

$$n(\text{Only }A)=n(A)-n(A\cap B)$$ $$n(\text{Only }B)=n(B)-n(A\cap B)$$

Use when the question asks for people belonging to exactly one category.

• Example: Out of 200 students, 120 study Python, 80 study Java, and 50 study both. How many study only Python? Solution:

       Python          Java
      _________      _________
     /         \____/         \
    /    70     50      30     \
    \___________/  \___________/

$$\text{Only Python}=120-50$$ $$\boxed{70}$$

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3. Maximum/Minimum Overlap

• Formula: For two sets:

$$\boxed{\text{Maximum overlap}=\min(n(A),n(B))}$$ $$\boxed{\text{Minimum overlap}=\max(0,n(A)+n(B)-n(U))}$$

Use when the exact intersection is not given but its possible range is required.

• Example: In a group of 100, 60 play cricket and 50 play football. Find the maximum and minimum number who play both. Solution:

Maximum:

$$\min(60,50)=\boxed{50}$$

Minimum:

$$\max(0,60+50-100) =\boxed{10}$$

Therefore:

$$\boxed{10\leq n(C\cap F)\leq50}$$

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4. At Least One / Neither

• Formula:

$$n(\text{At least one})=n(A\cup B)$$ $$n(\text{Neither})=n(U)-n(A\cup B)$$

• Example: In a class of 120 students, 70 know C++, 50 know Java, and 30 know both. Find the number who know neither. Solution:

$$n(C\cup J)=70+50-30=90$$ $$n(\text{Neither})=120-90$$ $$\boxed{30}$$

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5. De Morgan’s Laws & Complements

• Formula:

$$(A\cup B)'=A'\cap B'$$ $$(A\cap B)'=A'\cup B'$$

Therefore:

$$n(A'\cap B')=n(U)-n(A\cup B)$$

Use when the question contains words such as neither, not A and not B, or not both.

• Example: If (n(U)=100), (n(A)=40), (n(B)=50), and (n(A\cap B)=20), find (n(A’\cap B’)). Solution:

A' ∩ B'  =  Outside both A and B

$$n(A\cup B)=40+50-20=70$$

Hence:

$$n(A'\cap B')=100-70$$ $$\boxed{30}$$

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6. Complement of Intersection

• Formula:

$$n((A\cap B)')=n(U)-n(A\cap B)$$

Also:

$$(A\cap B)'=A'\cup B'$$

Use when the question asks for people who are not in both groups.

• Example: In a group of 200, 80 know Python and Java. How many do not know both? Solution:

$$n((P\cap J)')=200-80$$ $$\boxed{120}$$

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7. Set Cardinality Constraints — Three Sets

• Formula:

$$\begin{aligned} n(A\cup B\cup C)=& n(A)+n(B)+n(C)\ &-n(A\cap B)-n(B\cap C)-n(C\cap A)\ &+n(A\cap B\cap C) \end{aligned}$$

Use for three-category Venn diagram problems.

• Example: In a class of 80 students, 30 play Hockey, 40 Cricket, 25 Tennis. 15 play Hockey and Cricket, 10 Cricket and Tennis, 8 Tennis and Hockey, and 5 play all three. How many play at least one? Solution:

             Hockey
            _________
           /         \
          /     5     \
     ____/_____________\____
    /    \             /    \
   /      \           /      \
  / Cricket\_________/ Tennis \
  \        /         \        /
   \______/___________\______/

$$n(H\cup C\cup T) =30+40+25-15-10-8+5$$ $$=67$$ $$\boxed{67}$$

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8. Exactly Two Sets in a Three-Set Problem

• Formula: If (x=n(A\cap B\cap C)), then:

$$\text{Only }A\cap B=n(A\cap B)-x$$ $$\text{Only }B\cap C=n(B\cap C)-x$$ $$\text{Only }C\cap A=n(C\cap A)-x$$

Use when the question asks for exactly two categories.

• Example: 20 students study both Maths and Physics, 15 study Physics and Chemistry, and 12 study Chemistry and Maths. If 5 study all three, how many study exactly two subjects? Solution:

Maths & Physics only:

$$20-5=15$$

Physics & Chemistry only:

$$15-5=10$$

Chemistry & Maths only:

$$12-5=7$$

Total exactly two:

$$15+10+7=\boxed{32}$$

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9. Exactly One Set in a Three-Set Problem

• Formula:

$$\text{Only }A=n(A)-n(A\cap B)-n(A\cap C)+n(A\cap B\cap C)$$

Similarly for (B) and (C). Use when asking for members belonging to exactly one of three categories.

• Example: In a class, 50 study Maths, 40 Physics and 30 Chemistry. Pairwise intersections are 20, 15 and 10 respectively, while 5 study all three. Find the number studying exactly one subject. Solution:

Only Maths:

$$50-20-10+5=25$$

Only Physics:

$$40-20-15+5=10$$

Only Chemistry:

$$30-15-10+5=10$$

Exactly one:

$$25+10+10=\boxed{45}$$

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10. Three-Set Neither

• Formula:

$$n(\text{None})=n(U)-n(A\cup B\cup C)$$

• Example: In a group of 100, 60 like A, 50 like B, 40 like C; pairwise intersections are 20, 15, 10 and all three = 5. Find how many like none. Solution:

$$n(A\cup B\cup C) =60+50+40-20-15-10+5$$ $$=110$$

This exceeds the total (100), so the given data are inconsistent.

$$\boxed{\text{No valid solution}}$$

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11. Unknown Intersection from Union

• Formula:

$$n(A\cap B)=n(A)+n(B)-n(A\cup B)$$

Use when the union, rather than the intersection, is given.

• Example: In a group of 100 students, 65 play cricket, 55 play football, and 90 play at least one. Find how many play both. Solution:

$$90=65+55-n(C\cap F)$$ $$n(C\cap F)=120-90$$ $$\boxed{30}$$

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12. Union with Disjoint Sets

• Formula: If (A\cap B=\varnothing):

$$n(A\cup B)=n(A)+n(B)$$

Use when the categories cannot overlap.

• Example: In a group, 35 students are boys and 25 are girls. If these sets are disjoint, how many students are there? Solution:

$$n(B\cup G)=35+25$$ $$\boxed{60}$$

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Advanced Variants

13. Bounds for Three-Set Intersection

• Formula: For three sets, the exact bounds of (n(A\cap B\cap C)) depend on the given pairwise intersections and set sizes. A useful necessary bound is:

$$n(A\cap B\cap C)\leq\min[n(A\cap B),n(B\cap C),n(C\cap A)]$$

and:

$$n(A\cap B\cap C)\geq \max\left(0,, n(A\cap B)+n(B\cap C)+n(C\cap A)-n(A)-n(B)-n(C)+n(U) \right)$$

• Example: In a group of 100, (n(A)=60,n(B)=50,n(C)=40), with pairwise intersections (30,25,20). Find a possible minimum bound for all three. Solution:

$$x\geq 30+25+20-60-50-40+100$$ $$x\geq25$$

Also:

$$x\leq\min(30,25,20)=20$$

Since the lower bound exceeds the upper bound, the data are inconsistent.

$$\boxed{\text{No valid configuration exists}}$$

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14. Four-Set Inclusion-Exclusion

• Formula:

$$\begin{aligned} n(A\cup B\cup C\cup D)=& \sum n(A_i) -\sum n(A_i\cap A_j)\ &+\sum n(A_i\cap A_j\cap A_k) -n(A\cap B\cap C\cap D) \end{aligned}$$

Use only when a four-category problem explicitly provides the required intersections.

• Example: Four sets each contain 20 elements. If all pairwise intersections are zero, how many elements are in their union? Solution:

Since all sets are disjoint:

$$n(A\cup B\cup C\cup D) =20+20+20+20$$ $$\boxed{80}$$