Set Language, Notation & Basic Operations
- Universal Set ($U$ or $\xi$): The set containing all elements under consideration.
- Empty Set ($\emptyset$ or $\{\}$): The unique set containing no elements ($n(\emptyset) = 0$).
- Membership & Subsets:
• $x \in A$: $x$ is an element of set $A$.
• $A \subseteq B$: every element of $A$ is in $B$ ($A$ is a subset of $B$).
• $A \subset B$: $A$ is a proper subset of $B$ ($A \subseteq B$ and $A \ne B$). - Fundamental Operations:
• Union ($A \cup B$): $\{x \mid x \in A \text{ or } x \in B\}$ (all elements in either or both).
• Intersection ($A \cap B$): $\{x \mid x \in A \text{ and } x \in B\}$ (elements shared by both).
• Complement ($A'$): $\{x \in U \mid x \notin A\}$ (elements in the universe but outside $A$).
• Set Difference ($A \setminus B$ or $A \cap B'$): elements in $A$ that are not in $B$. - De Morgan's Laws: $$(A \cup B)' = A' \cap B'$$ $$(A \cap B)' = A' \cup B'$$
Example 1 · Criterion A
Listing Elements of Combined Set Operations
Let the universal set be $U = \{x \in \mathbb{Z} \mid 1 \le x \le 12\}$.
Let $A = \{\text{multiples of } 3\} = \{3, 6, 9, 12\}$.
Let $B = \{\text{factors of } 12\} = \{1, 2, 3, 4, 6, 12\}$.
Find: (a) $A \cap B$ (b) $A \cup B$ (c) $A'$ (d) $(A \cup B)'$
- (a) Intersection $A \cap B$: Common elements in both $A$ and $B$: $$A \cap B = \{3, 6, 12\}$$
- (b) Union $A \cup B$: Combine all unique elements: $$A \cup B = \{1, 2, 3, 4, 6, 9, 12\}$$
- (c) Complement $A'$: All elements in $U$ not in $A$: $$A' = \{1, 2, 4, 5, 7, 8, 10, 11\}$$
- (d) Outside the union $(A \cup B)'$: All elements in $U$ not in $A \cup B$: $$(A \cup B)' = \{5, 7, 8, 10, 11\}$$
Answers: (a) $\{3, 6, 12\}$; (b) $\{1, 2, 3, 4, 6, 9, 12\}$; (c) $\{1, 2, 4, 5, 7, 8, 10, 11\}$; (d) $\{5, 7, 8, 10, 11\}$.
2-Set Venn Diagrams & Inclusion-Exclusion Principle
- Two-Set Venn Structure: A Venn diagram partitions the universe $U$ into 4 mutually exclusive regions:
1. $A \cap B'$ (only in $A$)
2. $A \cap B$ (in both $A$ and $B$)
3. $A' \cap B$ (only in $B$)
4. $(A \cup B)'$ (outside both $A$ and $B$) - Principle of Inclusion-Exclusion (2 Sets): $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ Because adding $n(A)$ and $n(B)$ counts the intersection $n(A \cap B)$ twice, we subtract it once.
Example 2 · Criterion A & D
Two-Set Survey Problem Solving
In a cohort of $60$ students, $38$ take Chemistry, $34$ take Physics, and $8$ take neither subject.
(a) Find the number of students who take both Chemistry and Physics.
(b) Draw and populate a 2-set Venn diagram.
(c) A student is selected at random. Find the probability that the student takes only Chemistry.
- (a) Calculating intersection $n(C \cap P)$: Students taking at least one science: $$n(C \cup P) = n(U) - n(\text{neither}) = 60 - 8 = 52$$ Using Inclusion-Exclusion: $$n(C \cup P) = n(C) + n(P) - n(C \cap P)$$ $$52 = 38 + 34 - n(C \cap P) = 72 - n(C \cap P)$$ $$n(C \cap P) = 72 - 52 = \mathbf{20\text{ students}}$$
- (b) Populating regions: • Only Chemistry $= n(C) - n(C \cap P) = 38 - 20 = 18$ • Both subjects $= 20$ • Only Physics $= n(P) - n(C \cap P) = 34 - 20 = 14$ • Outside $= 8$ (Total check: $18 + 20 + 14 + 8 = 60$ ✓).
- (c) Probability of only Chemistry: $$P(\text{only Chemistry}) = \frac{n(C \cap P')}{n(U)} = \frac{18}{60} = \frac{3}{10} = \mathbf{0.30}$$
Answers: (a) $20$; (b) Regions: $18, 20, 14, 8$; (c) $\frac{3}{10}$ ($0.30$).
3-Set Venn Diagrams & Multi-Set Logic (Extended)
- Three-Set Structure: Partitions $U$ into 8 mutually exclusive regions:
• Innermost center: $A \cap B \cap C$ (all 3)
• 3 two-set pairwise intersections: $(A \cap B \cap C')$, $(A \cap C \cap B')$, $(B \cap C \cap A')$
• 3 exclusive single-set regions: only $A$, only $B$, only $C$
• Outermost region: $(A \cup B \cup C)'$ (none of the 3). - Principle of Inclusion-Exclusion (3 Sets): $$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$$
- Solving Strategy: Always populate from the inside out ($A \cap B \cap C = x$), subtracting from double intersections and then single circles.
Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$.
Let $P = \{2, 3, 5, 7\}$ (prime numbers) and $Q = \{1, 3, 5, 7, 9\}$ (odd numbers).
List the elements of: (a) $P \cap Q$ (b) $P \cup Q$ (c) $P' \cap Q$
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(a) $P \cap Q = \{3, 5, 7\}$ [A1]
(b) $P \cup Q = \{1, 2, 3, 5, 7, 9\}$ [A1]
(c) $P' = \{1, 4, 6, 8, 9, 10\}$; $P' \cap Q = \{1, 9\}$ [A1]
Given that $n(U) = 45$, $n(A) = 28$, $n(B) = 22$, and $n(A \cap B) = 12$:
(a) Find $n(A \cup B)$. [1 mark]
(b) Find $n(A \cap B')$ (only $A$). [1 mark]
(c) Find $n((A \cup B)')$ (outside both). [1 mark]
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(a) $n(A \cup B) = n(A) + n(B) - n(A \cap B) = 28 + 22 - 12 = \mathbf{38}$ [A1]
(b) $n(A \cap B') = n(A) - n(A \cap B) = 28 - 12 = \mathbf{16}$ [A1]
(c) $n((A \cup B)') = n(U) - n(A \cup B) = 45 - 38 = \mathbf{7}$ [A1]
In a sports club of $80$ members:
• $45$ play Tennis
• $38$ play Badminton
• $14$ play neither sport
(a) Find the number of members who play both sports. [2 marks]
(b) If a member is selected at random, find the probability that they play exactly one sport. [2 marks]
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(a) Intersection: [2 marks]
$n(T \cup B) = 80 - 14 = 66$ [M1]
$n(T \cap B) = n(T) + n(B) - n(T \cup B) = 45 + 38 - 66 = 83 - 66 = \mathbf{17\text{ members}}$ [A1]
(b) Exactly one sport: [2 marks]
Only Tennis $= 45 - 17 = 28$
Only Badminton $= 38 - 17 = 21$
Total exactly one sport $= 28 + 21 = 49$ [M1]
$P(\text{exactly one}) = \frac{49}{80} = \mathbf{0.6125}$ [A1]
Shade or describe using set notation each of the following Venn diagram regions:
(a) The region outside set $A$ but inside set $B$.
(b) The region representing elements in neither set $A$ nor set $B$.
(c) State which of De Morgan's laws applies to part (b).
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(a) $A' \cap B$ (or $B \setminus A$) [A1]
(b) $(A \cup B)'$ or $A' \cap B'$ [A1]
(c) De Morgan's Law: $(A \cup B)' = A' \cap B'$ [A1]
A survey of $100$ students regarding language study yields:
• $48$ study French ($F$), $40$ study Spanish ($S$), $36$ study German ($G$)
• $18$ study French and Spanish
• $14$ study Spanish and German
• $16$ study French and German
• $8$ study all three languages
(a) How many students study at least one of the three languages? [2 marks]
(b) How many students study none of these languages? [1 mark]
(c) How many students study only French? [2 marks]
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(a) Inclusion-Exclusion for 3 sets: [2 marks]
$n(F \cup S \cup G) = n(F) + n(S) + n(G) - n(F \cap S) - n(S \cap G) - n(F \cap G) + n(F \cap S \cap G)$
$= 48 + 40 + 36 - 18 - 14 - 16 + 8 = 124 - 48 + 8 = \mathbf{84\text{ students}}$ [A2]
(b) None of the languages: [1 mark]
$n(\text{none}) = 100 - 84 = \mathbf{16\text{ students}}$ [A1]
(c) Only French: [2 marks]
French and Spanish only $= 18 - 8 = 10$
French and German only $= 16 - 8 = 8$
Only French $= n(F) - [10 + 8 + 8] = 48 - 26 = \mathbf{22\text{ students}}$ [A2]
Verify De Morgan's Law $(A \cap B)' = A' \cup B'$ using truth tables or membership logic for any element $x \in U$.
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Step 1: Left hand side definition [1 mark]
$x \in (A \cap B)' \iff \neg(x \in A \cap B) \iff \neg(x \in A \land x \in B)$ [M1]
Step 2: Logical negation of conjunction [2 marks]
By logical negation of AND: $\neg(P \land Q) \equiv \neg P \lor \neg Q$
$\neg(x \in A \land x \in B) \iff x \notin A \lor x \notin B$ [R2]
Step 3: Conversion to set union [1 mark]
$x \notin A \lor x \notin B \iff x \in A' \lor x \in B' \iff x \in (A' \cup B')$.
Since both sides are logically equivalent for all $x$, $(A \cap B)' = A' \cup B'$. $\blacksquare$ [R1]
From the survey in Question 5 ($100$ students), a student is selected at random.
Given that the student studies Spanish, find the probability that they also study French:
$$P(F \mid S)$$
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Step 1: Formula definition [2 marks]
$P(F \mid S) = \frac{n(F \cap S)}{n(S)}$ [M2]
Step 2: Substitution and evaluation [2 marks]
From Question 5: $n(F \cap S) = 18$ and $n(S) = 40$.
$P(F \mid S) = \frac{18}{40} = \frac{9}{20} = \mathbf{0.45}$ [A2]
In a group of $50$ people, $32$ drink Coffee ($C$) and $28$ drink Tea ($T$). The number of people who drink neither is equal to half the number of people who drink both.
Find the number of people who drink both Coffee and Tea.
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Step 1: Define variable [1 mark]
Let $x$ be the number of people who drink both ($n(C \cap T) = x$).
Then $n(\text{neither}) = \frac{x}{2}$ (so $x$ is an even integer) [M1].
Step 2: Express regions [2 marks]
Only Coffee $= 32 - x$
Only Tea $= 28 - x$
Both $= x$
Neither $= 0.5x$ [A2]
Step 3: Total equation and solution [2 marks]
$(32 - x) + x + (28 - x) + 0.5x = 50$
$60 - 0.5x = 50 \implies 0.5x = 10 \implies x = \mathbf{20\text{ people}}$ [A2]
A marketing team interviews $150$ consumers.
• $95$ have heard of Brand A.
• $70$ have heard of Brand B.
(a) What is the minimum possible number of consumers who have heard of both brands? [2 marks]
(b) What is the maximum possible number of consumers who have heard of both brands? [2 marks]
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(a) Minimum overlap: [2 marks]
The minimum overlap occurs when $n(A \cup B)$ is maximized to the entire sample size $150$:
$n(A \cap B)_{\min} = n(A) + n(B) - 150 = 95 + 70 - 150 = 165 - 150 = \mathbf{15\text{ consumers}}$ [A2]
(b) Maximum overlap: [2 marks]
The maximum overlap occurs when the smaller set is a complete subset of the larger set ($B \subseteq A$):
$n(A \cap B)_{\max} = \min(n(A), n(B)) = \mathbf{70\text{ consumers}}$ [A2]
Investigate the total number of distinct subsets $S(n)$ of a set with $n$ elements:
• Set with $0$ elements $\emptyset \to \{\emptyset\} \implies 1$ subset.
• Set with $1$ element $\{a\} \to \{\emptyset, \{a\}\} \implies 2$ subsets.
• Set with $2$ elements $\{a, b\} \to \{\emptyset, \{a\}, \{b\}, \{a, b\}\} \implies 4$ subsets.
(a) List all subsets for a set with $3$ elements $\{a, b, c\}$. [2 marks]
(b) State the general rule for $S(n)$ in terms of $n$. [1 mark]
(c) A set has $256$ subsets. How many elements does it contain? [2 marks]
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(a) Subsets for 3 elements: [2 marks]
$\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}$ ($8$ subsets) [A2]
(b) General formula: [1 mark]
Every element has 2 independent choices (included or excluded):
$\mathbf{S(n) = 2^n}$ [A1]
(c) Finding $n$: [2 marks]
$2^n = 256 = 2^8 \implies n = \mathbf{8\text{ elements}}$ [A2]
Sets & Venn Diagrams Quick Reference
Set Notation Symbols
Set Operations
Inclusion-Exclusion Principle
De Morgan's Laws
Top 4 Venn Diagram Traps
1. Double Counting the Intersection
When $30$ like apples and $10$ like both apples and oranges, the number of people who like ONLY apples is $30 - 10 = 20$. Writing $30$ inside the apple-only crescent double counts the intersection!
2. Forgetting the Outside Boundary $(A \cup B)'$
Always verify that all numbers in all regions (including the outside region of the rectangle) sum up to the universal set total $n(U)$.
3. Confusing $\cup$ and $\cap$
$\cup$ looks like a "U" for Union (combining all). $\cap$ looks like an arch/bridge for Intersection (only the shared overlap).
4. Conditional Probability Denominator
For $P(A \mid B)$, the sample space shrinks to $B$. The denominator is $n(B)$, NOT $n(U)$.
Sets & Venn Diagrams Check
Answer these 5 questions to test set notation, Venn regions, and Inclusion-Exclusion.
If $n(U) = 50$, $n(A) = 30$, $n(B) = 25$, and $n(A \cap B) = 15$, what is $n(A \cup B)$?
Which set represents elements that belong to set $A$ but do NOT belong to set $B$?
According to De Morgan's Laws, $(A \cup B)'$ is identically equal to which expression?
In a survey of $100$ people, $70$ drink tea, $60$ drink coffee, and $10$ drink neither. What is $P(\text{drinks coffee} \mid \text{drinks tea})$?
How many distinct subsets does a set with $5$ elements contain?