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Notes On Topics Of Algebra

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Attached files contain the basic concept of Algebra topics.

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  1. SET A set is a well-defined collection of objects or elements. Each element in a set is unique. Usually but not necessarily a set is denoted by a capital letter e.g., A, B, . enclosed between brackets { } , denoted by small letters a, b, . Set of all small English alphabets Set of all positive integers less than or equal to 10 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Set of real numbers U, V etc. and the elements are x, y etc. For example: {x 00 < x < 00} The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers). The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by @(phi). The number of elements of a set A is denoted as n(A) and hence n(Ø) = 0 as it contains no element. Union of Sets Union of two or more sets is the best of all elements that belong to any of these sets. The symbol used for union of sets is 'U' i.e. AUB = Union of set A and set B = {x : x EA or x E B (or both)} e.g. If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8} then AUBUC = {1, 2, 3, 4, 5, 6 8}. Intersection of Sets It is the set of all of the elements, which are common to all the sets. The symbol used for intersection of sets is 'n'. i.e. AnB = {x : x EA and x E B} e.g. If A = {1, 2, 3, 4} an B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then AnBnc = {2} Remember that n(AUB) = n(A) + n(B) -(AnB). Difference of Two Sets The difference of set A to B denoted as A - B is the set of those elements that are in the set A but not in the set B i.e. A -B = {x : x EA and x A}. In general A-B*B-A e.g. If A = {a, b, c, d} and B = {b, c, d} then A -B = Subset of a Set The symbol used IS {a, d} and B -A— A set A is said to be a subset of the set B is each element of the set A is also the element of the set B. —n i.e. A
  2. Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A C DB. e.g. If A = {a, b, c, d} and B = Equality of Two Sets {b, c, d} then B DA or equivalently A B (i.e. A is a super set of B). —n —n Sets A and B are said to be equal if A B and B A and we write A = B. Universal Set As the name implies, it is a set with collection of all the elements and is usually denoted by U. e.g. set of real numbers R is a universal set whereas a set A = [x : x < 3} is not a universal set as it does not contain the set of real numbers x > 3. Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to (or AC) = complimentary set of A = {x : x > 3}. Hence we can that set. e.g. If A = {x : x < 3 then say that A U = U i.e. Union of a set and its complimentary is always the Universal set and A n f i.e. intersection of the set and its complimentary is always a void set. Some of the useful properties of operation on sets are as follows: (or (Ac)c) = A, Aeo = A and A.O = AVU = IJ and A•IJ = A An(BeC) = Au(B.C) = (AOB) (Acc) A.B = Äeb = and Illustration: If A = {a, b, c} and B = Solution: {b, c, d} then evaluate A u B, A n B, A - B and {x {x {x {x : x E Aorx€ B} : x EA and x B} : x E B and x A} = {d} Natural Numbers The numbers 1, 2, 3, 4 Thus N = {1, 45 Integers . are called natural numbers, their set is denoted by N.
  3. The numbers z. Thus 1 (or Z) . are called integers and the set is denoted by I or Including among set of integers are: Set of positive integers denoted by 1+ and consists of {1 2 3 Set of negative integers, denoted by 1- and consists of { ... , Set of non-negative integers {0, 1, 2,...} called as set of whole numbers Set of non-positive integers { Rational Numbers All numbers of the form p/q where p and q are integers and q * 0, are called rational numbers and their set is denoted by Q. Thus Q = {p/q : p,q E I and and HCF of p,q, is 1}. It may be noted that every integer is a rational number since it can be written as p/ 1. It may also be noted that all recurring decimals are rational numbers. e.g., p = 0.3 = 0.33333..... And 10p - p = 3 = > 9p = 3 = > p = 3/9 = > p = 1/3, which is a rational number. Irrational Numbers There are numbers which cannot be expressed in p/q form. These numbers are called irrational numbers and their set is denoted by QC (i.e. complementary set of Q) e.g. V 2, 1 + V 3, p etc. Irrational numbers cannot be expressed as recurring decimals. Real Numbers The complete set of rational and irrational numbers is the set of real numbers and is denoted by R. Thus R = Q u QC. It may be noted that N CI CQ CR. The real numbers can also be expressed in terms of position of a point on the real line. The real line is the number line where the position of a point relative to the origin (i.e. 0) represents a unique real number and vice versa. -3 -2 .2 Real N'..rnbet 2 3 All the numbers defined so far follow the order property i.e. if there are two numbers a and b then either a < b or a = bora > b. INTERVALS Intervals are basically subsets of R and are of very much importance in calculus as you will get to know shortly. If there are two numbers a, b E R such that a < b, we can define four
  4. types of intervals as follows: Open interval: (a, b) = {x : a < x < b} i.e. end points are not included Closed interval: [a, b] = {x : a S x S b} i.e. end points are also included. This is possible only when both a and b are are finite. Open-closed interval Closed-open interval The infinite intervals are defined as follows: (a, 00) [a, 00) (-00, b) (-00, b] — Intervals are particularly important in solving inequalities or in finding domains etc.
  5. INEQUALITIES The following are some very useful points to remember: either a < b or a -a > -b i.e. a < band c < d = > a a < b and number c V CER inequality sign reverses if both sides are multiplied by a + c mb if m < 0 0 < a < b = > ar < br if r > 0 and ar > br ifr 0 and equality holds fora = 1 (a+(1/2)) S -2 va > 0 and equality holds for a SINOSODIAL CURVE METHOD In order to solve inequalities of the form (P(x)/Q(x)) 2 0, (P(x)/Q(x)) S 0, where P(x) and Q(x) are polynomials, we use the following method: If Xl and (Xl < are two consecutive distinct roots of a polynomial equation, then within this interval the polynomial itself takes on values having the same sign. Now find all the roots of the polynomial equations P(x) = 0 and Q(x) = 0. Ignore the common roots and write — f(x) (((x-a1)(x-a2)(x-a3) ....(X-ßm))) where al, 02, . ., ßm are distinct real numbers. Then f(x) an, ßl, ß2, • = 0 for X = al, 02, . an and f(x) is not defined for x = ßl, ß2, ........, ßm. Apart from these (m + n) real numbers f(x) is either positive or negative. Now arrange al, 02, • , an, ßl, ß2, • ....., ßm in an increasing order say Cl, c2, c, c4, c, ........, Cmq-n. Plot them on the real line. An draw a curve starting from right of Cr-nq-n along the real line which alternately changes its position at these points. This curve is known as the sinosodial curve. c. C c, Illustration: Let f(x) positive or negative. Find the intervals where f(x) is
  6. Solution: Exercise The relevant sinosodial curve of the given function is -5 -2 -1 f(x) > OVXE (-5, -2)U (-1, 3) U (7, 00) and f(x) < OVXE (-00, -5) U -1) U (3, = (x2-3x+2)/(x2-1). Find the intervals where f(x) is negative. Let f(x) If ((x-1)(x-2)2(x-3)3)/((x-4)2(x-5)6) > 0, then find the values of x. LOGARITHM The expression logb a is meaningful for a > 0 and for either 0 < b < 1 or b > 1. - blogba loga b = logc b/logc a logb a = l/loga b provided both a and b are non-unity a: a: > 0 if b > 1 logb al > logb a2 ABSOLUTE VALUE Let x E R. Then the magnitude of x is called it's absolute value and is, in general, denoted by Ixl. Thus IXI can be defined as, IXI x,
  7. Note that x = 0 can be included either with positive values of x or with negative values of x. As we know all real numbers can be plotted on the real number line, IXI in fact represents the distance of number 'x' from the origin, measured along the number-line. Thus, IXI 2 0. Secondary, any point 'x' lying on the real number line will have its coordinate as (x, 0). Thus its distance from the origin is Vx2. Hence IXI = Vx2. Thus we can defined IXI as IXI = Vx2 or IXI = max (x, -x) e.g. if x = 2.5, then IXI = 2.5, if x = 3.8 then IXI = 3.8. Basic Properties of I xl = IXI IXI > a IXI < a = > x < a or x < -a if a E R+ and x E R if a E R- = > -a < x < a if a E R+ and no solution if a E R- u {0} Ix-y12 Iyl The last two properties can be put in one compact form namely, Iyl s + Iyl Ixyl = = Ix/yl 0 Illustration: Solution: Solve the inequality for real values of x: Ix -31 > 5. =>x< -2 or x > 8 = > XE (-00, -2) U (8, 00) GREATEST INTEGER AND FRACTIONAL PART Let x E R then [x] denotes the greatest integer less than or equal to x and {x} denotes the fractional part of x and is given by {x} = x - [x]. Note that 0 < {x} < 1. e.g. x = 2.69 =>2 -4 < x < -3 = > [x] It is obvious that if x is integer, then [x] = x. Basic Properties of greatest Integer and Fractional Part * = [00] = o, =
  8. n + [x] where n El 0, 1, if x E integer if x e integer x E integer x integer Hence [x] + [y] < [x + y] S [x] + [y] +1 = [x/n] , n E N, Illustration: If y Solution: XER - 3] + 5, then find the value of [x + y]. We are given that 3[x] + 1 = 2([x] -3) + 5 Hence [x + y] Illustration: Solve the equation = 3[x] + 2{x} for x. Solution: Case 1: For x < 1/2, -Il - Now 0 S {x} < 1 = > = 0 S 5x - 1 1/5 s x < 2/5 = > [x] = o = > x = {x} = > x = 5x-1 = > x = 1/4, which is a solution. Case 11: For > 1/2, 1 = > 2x - 2x-
  9. Now O < {x} < 1 = > o < x +1 1/2. Hence x = 1/4 is the only solution = > -1
  10. Cartesian Product Let A and B are two non-empty sets. The Cartesian product A x B of these sets is defined as the set f all ordered pairs (a, b) such that a EA and b E B. For example If A = {2, 3. 4} and B = {4, 9, 16} then A x B = 9), (2, 16), (3, 9), (3, 4), (3, 16), (4, 9), (4, 4), (4, 16)}. Out of these ordered paired elements of some are related with each other (In above example the second elements of a bold marked ordered pair is the square of the first. Such ordered pairs are called related ordered pairs. A subset 'f' of the Cartesian product AX B is called a function from A to B if and only if to each 'a' E A, there exists a unique 'b' in B such that (a, b) E f. Thus the function from A to B can be described as the set of ordered pairs (a, b) such that a E A and b E B and for each 'a' there is a unique 'b'. This function may be written as: Thus, a relation from A to B is a function if and only if to each a E A, there exists a unique 'b' n B such that (a, b) E f (al, bl) Ef and (alb2), Ef = > bl = b2. Graphically, if f(x) is plotted in the y axis against x and if a line parallel to y axis cuts f(x) at more than one point then f(x) does not fulfill the requirement of a function, because for same value of x, you will have two values of y. (see fig. 1) o 1 x +2. x = Y2 is not a function, because of x = e.g. If A = [a, b, c} and B = {d, e, f} Then f = { (a, d), (b, d), (c, e)} is a function from A to B but g = { (a, d), (a, e)} is not a function from A to B. The unique element b E B assigned to a E A is called the image of 'a' under f for value of f at 'a'. 'a' is called the pre-image of 'b'. Also 'a' is called theindependent variable, and 'b' is called the dependent variable.
  11. N/A

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