MTH 101 PDF Sets, Surds & Partial Fractions - Past Paper AQA, 2012

Okay, here is the markdown conversion of the images you sent: ## MAT 101 ### COURSE ONTLINE 1. Elementary set theory: Subset, Union Intercats, Complement, Venn diagrams 2. Real numbers: Integrab rational & lerational members, remainder theorem & partial fractions 3. Mathematical Induction; Rea...

Okay, here is the markdown conversion of the images you sent: ## MAT 101 ### COURSE ONTLINE 1. Elementary set theory: Subset, Union Intercats, Complement, Venn diagrams 2. Real numbers: Integrab rational & lerational members, remainder theorem & partial fractions 3. Mathematical Induction; Real Sequences & Series, theory of quadratic equations, Ranincal theorem 4. Complex numbers, algebra of complex numbers, fargand diagram, removal theorem, trigonometry, function of angle of any magnitudes, addition and factor formulae ### SET THEORY Set is a well-defined collection of objects e.g. litering of books, Kitchen utensils, maths 101 students. Set is usually denoted by a capital letter such as ‘A’. The member of a set. #### Set Notation and terminology **Element**: The object or members of a set are Called Ahementes $a \in A$ i.e. a belong/member of element of A **Subset**: A Set 'A' is said to be subset of a Set 'B'. If every element 'a' belonging to 'A' belongs to 'B' i.e. A is a Subset of B, denoted A ⊂ B **Proper subset**: * A Set 'A' is said to be proper subset of a Set 'B' IF AND IF ONLY IF; A is a Subset of B i.e. $A \subset B$ * If there exists at least an element of B that is not in A or there exists $b \in B$ Such that $b \notin A$ Ex2: * Let $A = {a, e}$ * $B = {a e, i, 0, 4}$ * i.e Every set is a sub-set of itself bout a proper subset * If A is a set, then $\phi \subset A$ **Complement of a set** $A^1 = {x : x \in U \text{ and } x \notin A}$ Ex 1: Write out the subset of $A = {1, 2}$, they are ${1}, {2}, {1, 2}, \phi, {\phi}$ N/B $= {0} \neq {\phi} = \Omicron \text{ is an element}$ If you have $n$ elements in a set then we have $p \in 2^n$ Subsets **Empty/Null and void set** A Set which contains no element is Called an empty, null or void set denoted by $\phi, { }$. e.g. A set of female governors in Nigeria is a null set. An even prime number greater than 2 is empty A singleton or wiset A set which Contains only one element is Called a UNISET e.g. Even prime number which is $[2]$ **Equality of a fet** Two set “A” & “B” are said to be equal if A = B i.e. AΩB **Difference of turs set** Let A and B be tuwe given sets then $A = B = {x: x \in A \text{ and } x \notin B}$ e.g. $A = {1,2,3, a, b, c, d}$ $B = {1, -5, 0, i, 2, i3, b, c}$ ∴ $A-B = {a,d}$ ∴ B-A $ = {-1, -5, 0}$ A – B ≠ B – A ∴ the difference btw the sets is not **Universal Set** This Contains every element we have inmimnd at a particular situation. A Set which Contains all other sets H's Subset. It's densted by V or $ \varepsilon $. **Complement of the Set** Let A be a subset of Universal Set V, the cet U-A = A’ or $A^c$ $A^1 = {x: x \in V \text{ and } x \notin A}$ I see a labelled venn diagram here for $A$ and $A^1$ enclosed by a rectangle with the label $ \varepsilon $ e.g. $V = {1, 2, 3, 4, 5, a, b, c, d, e}$ $A = {1, 2, 4, a, d}$ $A^1 = {3, 5, b, c, e}$ **Exercise 1** Which of the Option is different from the other and why & A) Empty for $\phi$ B) Natl C) Void D) Zero Ans IS D CO2 Zero is an element **Exercise 2:** Consider the following set $A = {a}, B = {a, c, b}, C = {c, a, b}, D = {c, b, a}, E = {b}, \phi$ Qnd. which of them is a subset of $x = {a, s, c}$? Ans. A, B, C, D, E, & are all subsets Qu2; which of them are proper sofs? Ans- A, C, E, & are proper subsets- **Exercise 3:** Consider the set below and answer the quest that follow $A = {\phi}, B = {3}, C = {1,5,9}, D = {1, 2, 3, 4, 5, 9}, E = {1, 3, 5, 7, 9},$ $ U = {1, 3, 4, a, -, 7, $, $\phi$ Insert the Correct Symbol C or ∉ for each plur I\) Q, A ⇒ QA II\) B, C ⇒ B ∉ C III\) B, E ⇒ BE N\) 6, D ⇒ C ∉ D V\) C, E ⇒ C ⊆ E VI\) D, E ⇒ D ∉ F VII\) D, U ⇒ D ⊆ U **Pover Sed**:- The set of all the Subsets of a set X is called the POWER SET of the (et X It is usually densted by $P(x)$ **Operation of Set** Union of Set : Let A and B be tisu sets the set that combine both A and B without repitition is Called the UNION OF A AND B l.e. $AUB = x: x \in A \text{ or } x \in B \text{ or } x \in A \text{ and } B$ I see a labelled Venn diagram here with two partially overlapping ovals with the labels A and B . e.g. Let $A = {a, b, c, d, e, f}$ $ B = {a, b. h, k}$ ⇒$(AUB) = {a, b, c, d, e, f, h, ky}$ $A = {x/x=y}$ $B = {x/3x+2=0 \text{ and } 20x+43=0}$ ∴ $AUB = {-2, 2, -2/3, -3/2}$ **Remark**: The notion of union of set Canbe extended to more them 2 sets le AUBUC (I) AUQ=A, AVA = A (A U B ) = (B V A) (Commutative) (AUB)UC = AV(BVC) (associative) **Intersection** Let A and B be sets, then the Intersection A and R equals **(13/9/12)** A ∩ B [Venn Diagram Image] **Disjoint Sets** - Let A god B are disjoints te. A ∩ B = $ \Omicron $ Point of Intersection between & tom setits. If there's d2 sets doult ay anifi in Commun Two sets are saud fobe dayout if they don't have any common element le A ∩ B = $ \Omicron $ or ${ }$ **Symmetric difference of tar sets** Let A and B be two sets, the Symmetric difference of A and B written as A ∆ B 15-- i.e. $A \Delta B = (A-B)U(B-A)$ [ labelled image with two partially overlapping sets labelled A and B] **Order of a sek** The Order of a finite set is the no of dement in that set eg. $A ={1, 2, 4}$ $n(A)=3$ **Exl** Proove that $A-B = A \cap B^!$ bay 1) Using Venn diggra, 10 by ngirous method see labelled Venn diagram of the set A and B $A \cap B^'$ $2: A - B \subset A \cap B^!$ $A \cap B^! \subset A - B$ Let $X$ belong to $A - B$ i.e $x \in A - B$ $\implies x \in A \text{and} X \notin B$ $\implies x \in A \text{ and } x E B$ **(13/2/12)** Let x belong to $A \cap B^!.$ i.e $x \in A \text{ and } x \notin B^!$ $\implies x \in A \text{ and } x \notin B$ i.e $x \in A - B$ $A \cap B^! \subset A - B$ $Hence A - B = A \cap B^!$ **Exercise:** Prove that $(AUB)^! = A^! \cap B^!$ Using (1) Venn diagrams VD Rigorous method see labelled Venn diagrams here of sets A, B, $A^!$,$ B^!$ Let $x \text { belong to } (AUB)^! $ $\implies X \in A^! \text{ and } X \in B^! $ $\implies X \in A^! B^!$ Let x belong to $A^! \cap B^! $ $\implies X \in A \text{ and } X \notin B^! $ (AUB)^! ASSIGNMENT PROVE ngorously = $AV(BNC) = (AUB)^!\cap (AUC)^!$ **(7/9/12) APPLICATION OF SET THEORY** **Example.lt** In Maths lol tuperial group of 30 students, 17 Students Studied Physical Sering 15 Mathematical De while 10 studied neithe mathematical nor physical; I) How many Students studied both? L II) Studied Mathematical Science only Solution [Venn Diagram Image] let a = 17 - x, b + x = 15 * b =15 17 – x + x + 15 – x + 10 = 30 32 + 10 – x = 30 ∴∴ 12 students studied bath. (ii) For mathematical Sc. mly 15−x x = 12 15−12=3 Students Studied mathematical Scene mly **Example 2:** Let $A, B, C$ be subsets of a universal set U defined by $ = A-(ARB)$ Draw on a Separate Venn diagrams I) $(A^! B)C$ II) $(A = C) B$ ( Assignment ) Soln $(A^! B) C^!$ Assume Let $(A^! B \text{ be }D)$ [Here onward, there are handwritten formulas and Venn diagrams to show that A B = A-(A ∩ B) See handwritten venn diagrams to describe further] A/16/12 **REAL NUMBERS** Integers 6 I numbers btw-Infinity to Infinity and His densted loy Z $Z = { - - - - -2, -1, 01:32 - - - }$ Natural numbers & of the Indigers blw I to Infinity and it's densted by N $N = {1, 2, 3}$ Rational numbers are numbers that can be written as a provided 'b' is not equal to o It's represented as Q $Q ={a/b, b \ 0 }$ Irrational numbers & numbers that con not be written as ? Ae.g vo, TR, To Real no A real ich kand and real line' **SURD** They & Irrational number and itts & root of of an anoth metic no whose value coulnt be obtained exauth. eng √2, 2√3 Different Surd can be combainced together by arithmetic Cooperation and that is called a SURD OPERATION A Surd dat Comprises of a Single term is Calleo monomial sound 0.2, √2, π√3 #### Rationalization of Surds **CONJUGATE SURD** Suppose me hace a binomial Sird “√3 +√2" of I form Then Conjugate of of Surd "is √3-√2 -> Rationalization of Surd is a means of Samplifying a Surd expression in the lowest or Simpliest from. Exl- Rationalize 3√2−5√3/√3−√2 Solution: => (3√2−\5√3/√3−√2) (√3+√2/√3+√2) => (3√2−\5√3/√3−√2) (√3+√2/√3+√2) => 20 ++√15+6\510/5-3 => 36+10/512 (see handwritten math from the images for more detailed derivations) **(27/9/12)** **SQUARE ROST'S OF SURTES** Suppuse We ax a Surd expressed we √a † √b and of Square root Square root is received of procedure Invsliced assuming! [see handwritten formulas] I've tried my best with all the handwritten maths and diagrams you included, please ask if you need clarifications.