MATHEMATICAL WORLD

  RELATIONS 1. INTRODUCTION : Let \(A\) and \(B\) be two sets. Then a relation \(R\) from \(A\) to \(B\) is a subset of \(A \times B\) . thus, \(R\) is a relation from \(A\) to \(B \Leftrightarrow R \subseteq A \times B\) . Total Number of Relations : Let \(A\) and \(B\) be two non-empty finite sets consisting of \(m\) and \(n\) elements respectively. Then \(A \times B\) consists of mn ordered pairs. So total number of subsets of \(A \times B\) is \(2^{m n}\) . Domain and Range of a relation : Let \(R\) be a relation from a set \(A\) to a set \(B\) . Then the set of all first components or coordinates of the ordered pairs belonging to \(R\) is called to domain of \(R\) , while the set of all second components or coordinates of the ordered pairs in \(R\) is called the range of \(R\) . Thus, \(\quad\) Domain \(( R )=\{ a :( a , b ) \in R \}\) and, Range \(( R )=\{ b :( a , b ) \in R \}\) It is evident from the definition that the domain of a relation from \(A\) to...

MATHEMATICAL REASONING 1. STATEMENT : A declarative sentence which is either true or false but not both, is called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is \(T\) and if it is false then its truth value is \(F\) . All Chapter Notes, Concept and Important Formula 2. SIMPLE STATEMENT: Any statement whose truth value does not depend on other statement are called simple statement. 3. COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement. Here the simple statements which form a compound statement are known as its sub statements. 4. LOGICAL CONNECTIVES : The words or phrases which combines simple statements to form a compound statement are called logical connectives. \(\scriptsize{ \begin{array}{|l|l|l|l|l|} \hline \text { S.N. } & \text { Connectives } & \text { Symbol } & \text { Use } ...

STATISTICS MEASURES OF CENTRAL TENDENCY: An average value or a central value of a distribution is the value of variable which is representative of the entire distribution, this representative value are called the measures of central tendency. Generally there are following five measures of central tendency: (a) Mathematical average (i) Arithmetic mean.           (ii) Geometric mean.           (iii) Harmonic mean (b) Positional average (i) Median.       (ii) Mode All Chapter Notes, Concept and Important Formula 1. ARITHMETIC MEAN/MEAN: (i) For ungrouped dist.: If \(x _{1}, x _{2}, \ldots \ldots x _{ n }\) are \(n\) values of variate \(x\) then their mean \(\bar{x}\) is defined as \(\overline{ x }=\dfrac{ x _{1}+ x _{2}+\ldots . .+ x _{ n }}{ n }=\frac{\displaystyle \sum_{ i =1}^{ n } x _{ i }}{ n }\) \(\Rightarrow\displaystyle \Sigma x _{ i }= n \overline{ x }\) (ii) For ungrouped and grouped freq. dist.:...

PROBABILITY 1. SOME BASIC TERMS AND CONCEPTS (a) An Experiment: An action or operation resulting in two or more outcomes is called an experiment. (b) Sample Space : The set of all possible outcomes of an experiment is called the sample space, denoted by S. An element of \(S\) is called a sample point. (c) Event : Any subset of sample space is an event. (d) Simple Event : An event is called a simple event if it is a singleton subset of the sample space \(S\) . (e) Compound Events : It is the joint occurrence of two or more simple events. (f) Equally Likely Events : A number of simple events are said to be equally likely if there is no reason for one event to occur in preference to any other event. (g) Exhaustive Events : All the possible outcomes taken together in which an experiment can result are said to be exhaustive. (h) Mutually Exclusive or Disjoint Events : If two events cannot occur simultaneously, then they are mutually exclusive. If \(A\) and \(B\) are mutually exclusive...

3D-COORDINATE GEOMETRY 1. DISTANCE FORMULA: The distance between two points \(A \left( x _{1}, y _{1}, z _{1}\right)\) and \(B \left( x _{2}, y _{2}, z _{2}\right)\) is given by \(A B=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}\right]}\) All Chapter Notes, Concept and Important Formula 2. SECTION FORMULAE : Let \(P \left( x _{1}, y _{1}, z _{1}\right)\) and \(Q \left( x _{2}, y _{2}, z _{2}\right)\) be two points and let \(R ( x , y , z )\) divide \(PQ\) in the ratio \(m _{1}: m _{2}\) . Then \(R\) is \((x, y, z)=\left(\frac{m_{1} x_{2}+m_{2} x_{1}}{m_{1}+m_{2}}, \frac{m_{1} y_{2}+m_{2} y_{1}}{m_{1}+m_{2}}, \frac{m_{1} z_{2}+m_{2} z_{1}}{m_{1}+m_{2}}\right)\) If \(\left( m _{1} / m _{2}\right)\) is positive, \(R\) divides \(PQ\) internally and if \(\left( m _{1} / m _{2}\right)\) is negative, then externally. Mid point of \(PQ\) is given by \(\left(\frac{ x _{1}+ x _{2}}{2}, \frac{ y _{1}+ y _{2}}{2}, \frac{ z _{1}+ z ...

  SETS 1. SET : A set is a collection of well defined objects which are distinct from each other. Set are generally denoted by capital letters \(A, B, C, \ldots .\) etc. and the elements of the set by a, b, c .... etc. If a is an element of a set \(A\) , then we write \(a \in A\) and say a belongs to \(A\) . If a does not belong to \(A\) then we write \(a \notin A\) , All Chapter Notes, Concept and Important Formula 2. SOME IMPORTANT NUMBER SETS : \(N =\) Set of all natural numbers \(=\{1,2,3,4, \ldots\}\) \(W =\) Set of all whole numbers \(=\{0,1,2,3, \ldots .\}\) \(Z\) or I set of all integers \(\{\ldots-3,-2,-1,0,1,2,3, \ldots\} \) \(Z ^{+}\) = Set of all +ve integers   \(=\{1,2,3, \ldots\}= N . \) \(Z ^{-}\) = Set of all -ve integers \(=(-1,-2,-3, \ldots .\}\) \(Z _{0}=\) The set of all non-zero integers . \(=\{\pm 1, \pm 2, \pm 3, \ldots\} \) Q = The set of all rational numbers.  \(=\left\{\frac{p}{q}: p, q \in I, q \neq 0\right\} \) \(R \) = the set of all ...

VECTORS 1. Vectors Physical quantities are broadly divided in two categories viz (a) Vector Quantities & (b) Scalar quantities. (a) Vector quantities: Any quantity, such as velocity, momentum, or force, that has both magnitude and direction and for which vector addition is defined and meaningful; is treated as vector quantities. (b) Scalar quantities: A quantity, such as mass, length, time, density or energy, that has size or magnitude but does not involve the concept of direction is called scalar quantity. All Chapter Notes, Concept and Important Formula 2. REPRESENTATION : Vectors are represented by directed straight line segment magnitude of \(\overrightarrow{ a }=|\overrightarrow{ a }|=\) length \(PQ\) direction of \(\overrightarrow{a}=P\) to \(Q\) 3. Types of Vectors (a) ZERO VECTOR OR NULL VECTOR : A vector of zero magnitude i.e. which has the same initial & terminal point is called a ZERO VECTOR. It is denoted by  \(\overrightarrow{O}\) . (b) UNIT VECTOR : A v...