Comprehensive Study Guide: Set Language, Notation, and Venn Diagrams

Starter Exercises: Weeks 9 and 10

Rectangle Perimeter and Area Bounds - Scenario: A rectangle has a length of $21\,cm$ (to the nearest $cm$ ) and a width of $5.3\,cm$ (to the nearest $mm$ ). - Identified Bounds: - Length Upper Bound ( $L(UB)$ ): $21 + 0.5 = 21.5\,cm$ - Length Lower Bound ( $L(LB)$ ): $21 - 0.5 = 20.5\,cm$ - Width Upper Bound ( $W(UB)$ ): $5.3 + 0.05 = 5.35\,cm$ - Width Lower Bound ( $W(LB)$ ): $5.3 - 0.05 = 5.25\,cm$ - Calculations: - Upper Bound for Perimeter ( $P(UB)$ ): $2 \times L(UB) + 2 \times W(UB) = 2 \times 21.5 + 2 \times 5.35 = 53.7\,cm$ - Lower Bound for Area ( $A(LB)$ ): $L(LB) \times W(LB) = 20.5 \times 5.25 = 107.625\,cm^2$ . - Final Answer (Area): Rounded to $2$ d.p., the answer is $107.63\,cm^2$ .
Sector Perimeter - Scenario: A sector $AOB$ of a circle with centre $O$ . Angle $AOB = 67^{\circ}$ . Radius $OA = OB = 5.2\,cm$ . - Arc Length Calculation: - Formula: $\frac{\theta}{360} \times 2 \times \pi \times r$ - Calculation: $\frac{67}{360} \times 2 \times \pi \times 5.2 = \frac{67}{360} \times 10.4 \times \pi$ - Intermediate value: $\frac{696.8 \times \pi}{360}$ - Total Perimeter Calculation: - Formula: $\text{Arc length} + 2 \times \text{radii}$ - Calculation: $(\frac{696.8 \times \pi}{360}) + 2 \times 5.2 = 6.08… + 10.4 = 16.5\,cm$ (correct to $3$ significant figures).

Fundamental Set Language and Keywords

Set: A collection of distinct objects, considered as an object in its own right, often defined using algebraic terms.
Element: An individual object or member of a set.
Subset: A set whose elements are all contained within another set.
Universal Set ( $\xi$ ): The set that contains all possible elements under consideration (denoted by the Greek letter xi).
Intersection ( $\cap$ ): The set containing all elements common to two or more sets (e.g., $\text{things in A AND B}$ ).
Union ( $\cup$ ): The set containing all elements in either of two or more sets (e.g., $\text{things in A OR B or BOTH}$ ).
Empty Set ( $\emptyset$ ): A set with no elements.
Cardinality ( $n(A)$ ): The number of elements in a set $A$ .
Set Characteristics: - Represented using curly braces: ${-4, 1, 3}$ . - Does not contain duplicates. - The order of elements does not matter (though usually written in ascending order).
Finite vs. Infinite Sets: - Finite Set: A set with a countable, limited number of elements (e.g., ${-4, 1, 3}$ ). - Infinite Set: A set that continues forever (e.g., the set of all positive integers or the set of all odd numbers).

Set Notation Symbols and Examples

$\in$ (Element of): Indicates a member of a set. Example: $2 \in {2, 4, 6, 8, 10, 12}$ .
$\notin$ (Not an element of): Indicates an object is not a member. Example: $3 \notin {2, 4, 6, 8, 10, 12}$ .
$\emptyset$ (Empty Set): If set $A = {\text{multiples of 4}}$ and $B = {\text{prime numbers}}$ , then $A \cap B = \emptyset$ (since $2$ is the only even prime and not a multiple of $4$ ).
$\xi$ (Universal Set): If $A = {3, 5, 7}$ and $B = {2, 4, 6, 8, 10}$ , a possible $\xi = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ .
$A'$ (Complement): Elements in the universal set that are NOT in set $A$ . - Example: If $\xi = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$ and $A = {3, 5, 7}$ , then $A' = {1, 2, 4, 6, 8, 9, 10}$ .

Practical Set Exercises and Solutions

Listing Elements: - Given $\xi = {\text{positive integers less than 20}}$ , $\xi = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}$ . - Set A (Prime numbers): $A = {2, 3, 5, 7, 11, 13, 17, 19}$ . - Set B (Square numbers): $B = {1, 4, 9, 16}$ . - Set C (Cube numbers): $C = {1, 8}$ . - Set D (Multiples of 4): $D = {4, 8, 12, 16}$ .
Operations on Sets (Integers less than 10): - $\xi = {1, 2, 3, 4, 5, 6, 7, 8, 9}$ . - $A = {1, 3, 7, 9}$ ; $B = {2, 3, 5, 6, 8, 9}$ . - $A'$ (Not A): $\xi - A = {2, 4, 5, 6, 8}$ . - $A \cup B$ (A or B): ${1, 2, 3, 5, 6, 7, 8, 9}$ . - $B'$ (Not B): $\xi - B = {1, 4, 7}$ . - $A \cap B$ (Overlap): ${3, 9}$ . - $A' \cap B$ (In B but not in A): ${2, 5, 6, 8}$ . (Transcript answer notes combined as $\text{Not A overlapped with B}$ ). - $A \cap B'$ (In A but not in B): ${1, 7}$ . - $A' \cup B'$ (Not A or Not B): ${1, 2, 4, 5, 6, 7, 8}$ .
True/False Checks: - $4 \in {1, 3, 4, 7}$ : TRUE (4 is a member). - $2 \notin {\text{prime numbers}}$ : FALSE (2 is a prime number). - $25 \notin {\text{multiples of 3}}$ : TRUE (25 is not divisible by 3).

Venn Diagrams and Visual Notation

Structure: - Use a rectangle to denote the Universal Set ( $\xi$ ). - Use circles to represent individual sets. - Circular regions must be labeled (crucial for marks in exams).
Visual Regions: - Intersection Only: Overlap of circles ( $A \cap B$ ). Items belonging to both sets. - Union: Everything inside both circles ( $A \cup B$ ). Items in either or both. - Complement: Region inside the rectangle but outside a circle ( $A'$ ). Items in $\xi$ not in $A$ . - Specific Subtraction: Items in $A$ but not in $B$ ( $A \cap B'$ ).
Drawing Procedure: 1. Determine if sets intersect ( $A \cap B \neq \emptyset$ ). 2. Place intersection elements first in the overlapping region. 3. Fill in the remaining elements of individual sets in their respective circles. 4. Place elements of $\xi$ that are not in either set outside the circles but inside the rectangle.

Starter Exercises: Weeks 11 and 12

Estimated Mean for Grouped Data: - Data: Distance travelled ( $d\,km$ ) to deliver $100$ parcels. - Table Data: - Boundaries: 0 < d \le 5; Frequency: $28$ ; Midpoint: $2.5$ ; $MP \times F = 70$ - Boundaries: 5 < d \le 10; Frequency: $32$ ; Midpoint: $7.5$ ; $MP \times F = 240$ - Boundaries: 10 < d \le 15; Frequency: $20$ ; Midpoint: $12.5$ ; $MP \times F = 250$ - Boundaries: 15 < d \le 20; Frequency: $14$ ; Midpoint: $17.5$ ; $MP \times F = 245$ - Boundaries: 20 < d \le 25; Frequency: $6$ ; Midpoint: $22.5$ ; $MP \times F = 135$ - Total Frequency (n): $100$ - Total Sum ( $\sum MP \times F$ ): $940$ - Estimated Mean: $\frac{940}{100} = 9.4\,km$
Triangle Angle Calculation: - Setup: Triangle $ABC$ . $ADB$ is a straight line. $\angle ABC = 83^{\circ}$ . $\angle BAC = 52^{\circ}$ . Line $CD$ cuts $\angle BCD$ . - Ratio: $\text{Size of } \angle BCD : \text{size of } \angle ACD = 2 : 3$ . - Steps: 1. Find total angle $\angle C$ : $180 - 83 - 52 = 45^{\circ}$ . 2. Sum of ratio shares: $2 + 3 = 5$ shares. 3. Value of one share: $45 / 5 = 9^{\circ}$ . 4. $\angle ACD = 9 \times 3 = 27^{\circ}$ . 5. Solve for $\angle ADC$ : In triangle $ADC$ , $\angle ADC = 180 - 27 - 52 = 101^{\circ}$ .

Probability Applications of Venn Diagrams

School Subject Example: - Total Year 11 students: $100$ . - History ( $H$ ): $50$ students. - Music ( $M$ ): $30$ students. - Both ( $H \cap M$ ): $20$ students. - Venn Breakdown: - Only History: $50 - 20 = 30$ - Both: $20$ - Only Music: $30 - 20 = 10$ - Neither: $100 - (30+20+10) = 40$ - Probabilities: - Not studying Music OR History: $\frac{40}{100} = 0.4$ - Only Music and NOT History: $\frac{10}{100} = 0.1$
Sports Example (75 Boys): - Football ( $F$ ), Rugby ( $R$ ), Cricket ( $C$ ). - Probabilities from random selection: - Played Football: $\frac{20}{75}$ - Played at least one sport: $\frac{40}{75}$ - Played Rugby but not Cricket: $\frac{11}{75}$ - Didn't play Football: $\frac{55}{75}$ - Played more than one sport: $\frac{8}{20}$ (Conditional probability: Played another sport given football). - Played Cricket given didn't play Rugby: $\frac{12}{59}$ . - Provided solutions list detailed fractions for these events, e.g., $\frac{11}{16}$ for final subset calculation.
Holiday Example (200 People): - Europe ( $E$ ), Asia ( $A$ ), USA ( $U$ ). - Probabilities: - Had been to Europe: $\frac{152}{200}$ - Exactly one place: $\frac{132}{200}$ - None of the three: $\frac{15}{200}$ - Europe but not Asia: $\frac{135}{200}$ - Europe and Asia given USA: $\frac{4}{60}$

Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Prime Factorization via Venn Diagrams: - Intersection of prime factor sets = HCF. - Union of prime factor sets (product of all numbers in the Venn diagram) = LCM.
Example 1: 12 and 20: - $12 = 2 \times 2 \times 3$ - $20 = 2 \times 2 \times 5$ - Overlap (common factors): $2, 2$ . - $HCF = 2 \times 2 = 4$ . - $LCM = 3 \times 2 \times 2 \times 5 = 60$ .
Example 2: 96 and 120: - $96 = 2^5 \times 3$ ( $2 \times 2 \times 2 \times 2 \times 2 \times 3$ ) - $120 = 2^3 \times 3 \times 5$ ( $2 \times 2 \times 2 \times 3 \times 5$ ) - Intersection: $2, 2, 2, 3$ . - $HCF = 2^3 \times 3 = 8 \times 3 = 24$ .
Example 3: Algebraic/Power Forms: - $A = 2^3 \times 5 \times 7^2 \times 11$ - $B = 2^4 \times 7 \times 11$ - $C = 3 \times 5^2$ - LCM of A, B, and C: Collect the highest power of every prime appearing in the factors. - $LCM = 2^4 \times 3 \times 5^2 \times 7^2 \times 11 = 646,800$ .

Exam Style Questions & Elaborations

Case Study: $\xi = {\text{integers } y \text{ s.t. } 3 \le y \le 25}$ - $R = {\text{even numbers in range}} = {4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}$ . - $S = {\text{integers } y \text{ s.t. } 12 \le y \le 22} = {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}$ . - $n(R)$ : There are $11$ elements. - $(R \cup S)'$ : Elements in $\xi$ but neither in $R$ nor $S$ . These are $\text{odd numbers not between 12-22}$ , specifically: ${3, 5, 7, 9, 11, 23, 25}$ . - $R' \cap S$ : Odd numbers within the set $S$ : ${13, 15, 17, 19, 21}$ .
Logical Explanation for $A \cap B = \emptyset$ : - Accepted exam explanations: "No members in common," "No overlap," or " $A$ has even numbers and $B$ has odd numbers except $2$ , which is not in $B$ ." - Specifically: If $A = {2, 3, 5, 7}$ and $B = {4, 6, 8, 10}$ , they share no common elements, hence the intersection is the empty set.
Disproving $\emptyset$ : - If student Taylor claims $N \cap P = \emptyset$ where $N = {2,3,4,5,6}$ and $P = {4,5,6,8,10}$ , the statement is false because the intersection is actually $\text{not empty}$ ; it contains \s{4, 5, 6}.