Mathematics DIctionary

Every student needs a mathematical dictionary suited to the classes of every level i.e. from School through Graduation. Find the below definitions used in day to day math.

Sets : A set is a well defined collection of distinct objects.E.g.: {x, y, z, …}. Types of setsFinite setA set consisting of a definite number of elements is called a finite set.E.g.:- {4,5,6,7,9}
Infinite setA set having an infinite number of elements is called an infinite number of elements is called an infinite set.E.g.: {0,1,2,3,…….}. Null setA set which has no element is a null set, which is denoted by f.E.g.: f = the set of integers between 6 and 7. 
Singleton setA set having only one element is called a singleton set.E.g.: A = {2}. Equal setsTwo sets A and B are said to be equal if they consist of exactly the same elements i.e. A=B.
Equivalent setsEquivalent sets are those which have the same number of elements. SubsetIf A and B are two sets such that every element of A is also an element of B, then A is a subset of B.  It is denoted by A Ì B.
Proper SubsetIf every element of A is in B but every element of B is not in A, then A is a proper subset of B. Union of two setsThe union of two sets A and B is the set consisting of all elements of A together with all the elements of B without repeating the elements more than once.I.e. A È B = {x/x Î A or x Î B}.
Intersection of two setsThe intersection of two sets A and B is the set of elements common to A and B.i.e. A Ç B = {x/x Î A and x ÎB}. Disjoint setsWhen two sets have no common element, they are called disjoint sets.i.e. A Ç B = f
De Morgan’s LawsIf A and B are two subsets of u, then(i) (A È B)C = AC Ç BC(ii) (A Ç B)C = AC È BC

 

Cartesian product of sets [Cross product]The set of all ordered pairs (a,b) where aÎA and bÎB is called the Cartesian Product of the two Sets A and B and is denoted by AxB.ie, A x B = {(a,b) / aÎA, bÎB}.
Domain and rangeA relation R from a set A to a set B is a subset of A x B.  Here A is called the domain of R. The set of second entries of the ordered pairs in a relation is called the range of the relation. Inverse relationEvery relation R from A to B has an inverse relation R-1 from B to A which is defined by,R-1 = {(b.a) / (a,b) ÎR}.
Reflexive relationsA relation R in a set A is said to be reflexive if a Ra for all a Î A.E.g.: Set of natural numbers. Symmetric relations            A relation R in a set A is said to be symmetric if a Rb Þ b Rafor all  a, b Î A.
Transitive relationsA relation R in a set A is said to be transitive if aRb, bRc Þ aRcfor all a,b,c Î A. Equivalence relationA relation R in a set A is said to be an equivalence relation if it is(i) reflexive(ii) symmetric, and

(iii) transitive

FunctionA function ‘f’ from X to Y is a subset of X X Y in which an ordered pair (x,y) Î X XY, xÎX, yÎY occurs only one time with x as the first element.ie, (x,y) Î ¦, (x,y2) Î ¦ Þ y1 = y2. Different types of functionsINTO and ONTO functions Let ¦:A®B be a function.  Then A = dom ¦ and B = co-dom ¦.  If the range, ran ¦, of ¦ is such that ran ¦ ÌB, ¦ is said to be a function defined on A into B.

If ran ¦ = B, ¦ is said to be a function defined on A onto B.  This function is also called Subjective function

ONE-ONE and MANY-ONE FunctionsLet ¦:A®B be a function.  Then ¦ is called a one-one or injective function if distinct elements of A are taken to distinct elements of B by ¦.Let ¦:A®B be a function.  Then ¦ is called a many-one function if more than one element in A have the same image in B. Bijective FunctionA function which is both one-one and onto is said to be a bijective function.
Constant FunctionLet ¦:A®B be a function, then ¦ is called a constant function if the range of ¦ consists of only one element. Identity FunctionLet A be any set.  Then the function ¦:A®A defined by ¦(a) = a, for all aÎA, is called the identity function of A.  It is denoted by IA.  Identity function is a bijective function.
Complex numbersA number of the form a+ib where a and b are real numbers is called a complex number.  The set of all complex numbers is denoted by C.\f 0 C = {a+ib / a,bÎR and i = Ö-1}. Conjugate of a complex numberIf Z = a+ib, then a+i(-b) = a-ib, denoted by Z is called the conjugate of Z.
Modulus of a complex number:If Z Î C, Z = a+ib; a,bÎR, then the non-negative square root of a2+b2, denoted by |Z|, is called the modulus or absolute value of the complex number Z.\f0  |Z| = Öa2+b2. De Moivre’s formula:For all integral values of n, positive, zero and negative.(cosq + i sinq)n = Cos nq + i Sin nq.
Symmetric functions of the RootsLet µ, b be the roots of the quadratic equation ax2+bx+c = 0, a¹0, then the relation between µ and b is said to be symmetric if it remains unaltered on interchanging µ and b. SequenceA Set of numbers occurring in a definite order or formed accordingly to some definite rule is called a sequence.
SeriesA sequence of numbers, connected together by the sign of addition is called a Series. Arithmetic progression (A.P).A finite or infinite sequence a1,a2,……,an is called an arithmetic progression if a –a K -1 = d, a constant independent of k for k = 2,3,……,n.  The nth term of an A.P whose first term is a1 and common difference d is,A n = a 1 + (n-1)d.The sum of n terms of an A.P where ‘Sn‘ denote the first n terms of an A.P, the first term ‘a’ and the common difference ‘d’ is,

n = n/2 [1st term + nth term].

Arithmetic means (A.M.s)In an Arithmetic progression of n terms, the terms between the first and the last are called the Arithmetic Means between them. Properties of conjugatesIf Z1 and Z2 are any two complex numbers, then(1) z1+z2 = z1+z2(2) z1z2 = z1 . z2

(3) z1z2 = z1z2

(4) (z1 / z2) = z1 / z2z2 ¹0

(5) (z) = z

 

Properties of modulusIf Z1 and Z2 are any two complex numbers, then(1) |Z1Z2| = |Z1| . |Z2|(2) Zz= |Z|2.

(3) |Z1/Z2| = |Z1| / |Z2|

(4) |Z1+Z2| £ |Z1| + |Z2|

(5) ||Z1| – |Z2|| £ |Z1-Z2|

 

Quadratic polynomial and quadratic equationA polynomial of the second degree is called a quadratic polynomial.  Any equation ¦(x) = 0 where ¦ is a 0quadratic polynomial, is called a quadratic equation.The general form of a quadratic equation is,ax2+bx+c = 0, where a, b, c are real numbers.

 

Equation and identityAn algebraic expression equated to another expression or zero is called an equation.An identity is a statement of equality between two expressions. Geometric means (G.Ms)In a finite sequence in G.P, the terms between the first and the last are called the geometric means between them.
Distance FormulaThe distance between two points P(x1,y1) and Q(x2,y2) is,d = |PQ| = Ö(x2-x1)2 + (y2-y1)2 Centroid of a triangleCentroid of a triangle = ((x1+x2+x3)/3, (y1+y2+y3)/3))
General equation of the second degreeEquation of the second degree is given by,ax2+2hxy+by2+2gx+2fy+c = 0. CircleA circle is the locus of a point which moves in a plane in such a way that its’ distance from a fixed point is always a positive constant.If the centre of the circle is at c(h,k) and has radius r, then the equation of the circle is given by,(x-h)2 + (y-k)2 = r2.
Equation of a circle when the end points of a diameter are given(x-x1) (x-x2) + (y-y1) (y-y2)= 0. Family of concentric circlesA family of circles with the same centre and different radii is called a family of concentric circles.  If g and f are known and k is a parameter, thenX2+Y2+2gX+2fY+k = 0 represents a family of concentric circles having (-g,-f) as centre.
Orthogonal circlesTwo circles are said to be orthogonal if the angle of their intersection is a right angle. Conic sectionsThe curves obtained by slicing a cone with a plane that does not pass through the vertex are called conic sections.
ParabolaA parabola is the set of all points whose distance from a fixed point in the plane are equal to their distances from a fixed line in the plane.  Standard equation of the parabola is y2 = 4ax. EllipseAn ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant ratio, less than one, to their distances from a fixed line in the plane.Standard equation of an ellipse is x2/a2 + y2/b2 = 1.
Slope of a lineThe slope m of a line having inclination µ and not perpendicular to x-axis, is defined to be tan µ. Locus and equationsWhen a point moves subject to a set of specified conditions, the path traced out by it is called its locus.Equation to a locus is the algebraic relation that exist between the x and y co-ordinates of a general point on the locus.
The Point-Slope formEquation of a straight line passing through the point (x1,y1) and having the slope m is,y-y1=m(x-x1). Slope-intercept formEquation of a straight line which cuts off a given intercept c on the y-axis and having slope m is,y=mx+c.
Normal formEquation of a straight line in terms of ‘P’, the length of the perpendicular from the origin upon it and ‘w’, the angle which this perpendicular makes with x-axis is,              x Cos  w + y sin w = P. Symmetric formEquation of a straight line passing through a given point (x1,y1) and making an angle q with the x-axis is,(x-x1)/Cosq = (y-y1)/Sinq = r.
Angle between two linesThe positive angle q from the line l1 to the line l2 with slopes m1 = tan µ1 and m2 = tan µ2 respectively is given by the equation,tanq = (m2-m1)/(1+m1m2)where l1 and l2 are two non-perpendicular lines. Translation of axesShifting the origin of a coordinate system without changing the direction of the axes is known as translation of axes.
Parametric equations(a) Parabola y2 = 4 ax.x = at2, y = 2at, t is the parameter(b) Ellipse x2/a2 + y2/b2=1

           x =  a cosq, y = a sinq.

Where q is called eccentric angle.

(c) Hyperbola x2/a2 – y2/b2 =1

x = a Secq, y=b tanq.

 

RadianA radian is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.Pythagorean relations(i) Cos2q + Sin2q = 1

(ii) tan2q + 1 = Sec2q

(iii) 1 + Cot2q = Cosec2q.

Sum and product formulae

Sin C + Sin D = 2 Sin (C+D)/2. Cos (C-D)/2.

Sin C – Sin D = 2 Cos (C+D)/2. Sin (C-D)/2.

Cos C + Cos D = 2 Cos (C+D)/2. Cos (C-D)/2.

Cos D – Cos C = 2 Sin (C+D)/2. Sin (D-C)/2.

Trignometric equationsAn equation involving one or more trigonometric functions of unknown angles is called a trigonometric equation. The Law of Cosinesa2 = b2 + c2 – 2bc Cos Ab2 = c2 + a2 – 2ca Cos Bc2 = a2 + b2 – 2ab Cos C

 

The area of a triangleThe area D of a triangle ABC is given by,D = ½  ab Sin C. Hero’s FormulaD = Ös(s-a)(s-b)(s-c). where s=(a+b+c)/2
Projection formulaea = b Cos C + c Cos Bb = c Cos A + a Cos Cc = a cos B + b Cos A.

 

Raw dataThe information collected through census and surveys is called raw data.Range of raw dataThe difference between the maximum and the minimum number occurring in the data is called the range of data.
Quantitative and qualitative variablesThe variables of observation with numbers as possible values are called quantitative variables. E.g.: Height, area of a plot of land etc.The variables with names of things, places, attributes etc as possible values are called qualitative variables.E.g.: Religion, Caste etc.

 

Continuous and discrete variablesA quantity which can take any numerical value within a certain range is called a continuous variable. E.g.: Pressure, temperature etc.A quantity which is incapable of taking all possible values is called a discrete variable. E.g.: Marks obtained by a student, number of members in a family etc.
Relative frequencyThe frequency expressed as a fraction of the total frequency and the fraction so obtained, usually expressed as a percentage, is called the relative frequency of the class. Cumulative frequency tableThe cumulative frequency of any class is the total of the frequencies of that class and all classes coming before it in the frequency table.  The table showing the manner in which cumulative frequencies are distributed is called cumulative frequency table.
The Binomial TheoremFor any positive integer n,(a+b)n = C(n,0)an + C(n,1) an-1b + C(n,2)an-2b2 +……..+ C(n,r)an-r br+…+C(n,n-1)  a bn-1 + C(n,n)bn.

 

where C(n,r) = (n!)/(r!(n-r)!) for 1£r£n.

Binomial Theorem for a rational index(1+x)n = 1+nx + n(n-1)/(1.2) x2+(n(n-1)(n-2))/(1.2.3) x3 +……………..for any rational number n and for |x| < 1.
Linear programmingLinear Programming is a mathematical method which attempts to maximise or minimise objectives. Resultsex = 1+x/1! + x2/2! + …………ex+y = ex . ey.

eiq = 1+ (iq)/(1!) – (q2)/(2!) – i (q3)/(3!) + (q4)/(4!) + ………..

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