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Mathematics | PnC and Binomial Coefficients Several Counting problems require finding the number of ways to arrange a certain number of distinct elements, where the relative order of these elements matter, other problems focus on finding the number of ways of selecting a particular number of elements from a set, where the order of the elements does not matter. Both types of problems are similar except for one crucial difference, that difference is order. Permutation – A permutation of a set of distinct objects is an ordered arrangement of these objects. A permutation is often also referred to as an arrangement. The relative order of the elements matters in an arrangement. An ordered arrangement of elements of a set is called an r-permutation. • Example 1 – How many permutations of the string “ABCDEFGH” have the string “ABC” as a substring? • Solution – For “ABC” to be a substring, the letters A,B, and C must occur as a block. If we consider that block and the remaining 5 letters as objects, we have a total of 6 objects to arrange. Therefore the number of strings having “ABC” as their substring = 6! = 720. Combination – A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. The order of elements does not matter in a combination. An unordered selection of elements from a set is called an r-combination. Binomial Coefficients – The -combinations from a set of n elements. This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions. The binomial theorem gives a power of a binomial expression as a sum of terms involving binomial coefficients.

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Rolle’s Mean Value Theorem Suppose f(x) be a function satisfying three conditions: 1) f (x) is continuous in the closed interval a ≤ x ≤ b 2) f (x) is differentiable in the open interval a < x < b 3) f (a) = f(b) Then according to Rolle’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that: f ‘ (c) = 0 So, we can apply Rolle’s Theorem, according to which there exists at least one point ‘c’ such that: f ‘ (c) = 0 Which means that there exists a point at which the slope of the tangent at that is equal to 0. We can easily see that at point ‘c’ slope is 0. Similarly, there could be more than one points at which slope of tangent at those points will be 0.

Mathematics | Mean, Variance and Standard Deviation Mean is average of a given set of data. Let us consider below example 2, 4, 4, 4, 5, 5, 7, 9 These eight data points have the mean (average) of 5: (2 + 4 +4 + 4 +5 +5 +7 + 9)/8 = 5 Variance is the sum of squares of differences between all numbers and means. Deviation for above example. First, calculate the deviations of each data point from the mean, and square the result of each: Standard Deviation is square root of variance. It is a measure of the extent to which data varies from the mean. Standard Deviation (for above data) = √4 = 2 Why mathematicians did chose a square and then square root to find deviation, why not simply take the difference of values? One reason is the sum of differences becomes 0 according to the definition of mean. Sum of absolute differences could be an option, but with absolute differences, it was difficult to prove many nice theorems. Coefficient of Variation = Standard Deviation/Mean * 100 * Value of standard deviation is 0 if all entries in input are same. * If we add (or subtract) a number say 7 to all values in the input set, the mean is increased (or decreased) by 7, but the standard deviation doesn’t change. * If we multiply all values in the input set by a number 7, both mean and the standard deviation is multiplied by 7. But if we multiply all input values with a negative number say -7, the mean is multiplied by -7, but the standard deviation is multiplied by 7. * Standard deviation and variance is a measure that tells how spread out the numbers is. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean. * Mean, median and mode are the measure of central tendency of data (either grouped or ungrouped).

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Mathematics | Probability Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from deck, etc., then the must be some probability associated with that event. In terms of mathematics, probability refers to the ratio of wanted outcomes to the total number of possible outcomes. There are three approaches to the theory of probability, namely: 1. Empirical Approach 2. Classical Approach 3. Axiomatic Approach Basic Terminologies: • Random Event :- If the repetition of an experiment occurs several times under similar conditions, if it does not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, then such an experiment is called random event or a probabilistic event. • Elementary Event – The elementary event refers to the outcome of each random event performed. Whenever the random event is performed, each associated outcome is known as elementary event. • Sample Space – Sample Space refers to the set of all possible outcomes of a random event. Example, when a coin is tossed, the possible outcomes are head and tail. • Event – An event refers to the subset of the sample space associated with a random event. • Occurrence of an Event – An event associated with a random event is said to occur if any one of the elementary event belonging to it is an outcome. • Sure Event – An event associated with a random event is said to be sure event if it always occurs whenever the random event is performed. • Impossible Event – An event associated with a random event is said to be impossible event if it never occurs whenever the random event is performed. • Compound Event – An event associated with a random event is said to be compound event if it is the disjoint union of two or more elementary events. • Mutually Exclusive Events – Two or more events associated with a random event are said to be mutually exclusive events if any one of the event occurs, it prevents the occurrence of all other events. This means that no two or more events can occur simultaneously at the same time. • Exhaustive Events – Two or more events associated with a random event are said to be exhaustive events if their union is the sample space. Probability of an Event – If there are total p possible outcomes associated with a random experiment and q of them are favorable outcomes to the event A, then the probability of event A is denoted by P(A) and is given by P (A) = q/p The probability of non-occurrence of event A, i.e. P(A’) = 1 – P(A) Note – • If the value of P (A) = 1, then event A is called sure event. • If the value of P (A) = 0, then event A is called impossible event. • Also, P(A) + P(A’) = 1 Theorems: • General – Let A, B, C are the events associated with a random experiment, then 1. P(A∪B) = P(A) + P(B) – P(A∩B) 2. P(A∪B) = P(A) + P(B) if A and B are mutually exclusive 3. P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) – P(B∩C)- P(C∩A) + P(A∩B∩C) 4. P(A∩B’) = P(A) – P(A∩B) 5. P(A’∩B) = P(B) – P(A∩B) • Extension of Multiplication Theorem – Let A1, A2, ….., An are n events associated with a random experiment, then • P (A1∩A2∩A3 ….. An) = P (A1) P (A2/A1) P (A3/A2∩A1) ….. P (An/A1∩A2∩A3∩ ….. ∩An-1)

Consensus Theorem in Digital Logic Redundancy theorem is used as a Boolean algebra trick in Digital Electronics. It is also known as Consensus Theorem: AB + A'C + BC = AB + A'C The consensus or resolvent of the terms AB and A’C is BC. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. The conjunctive dual of this equation is: (A+B). (A'+C). (B+C) = (A+B). (A'+C) In the second line, we omit the third product term BC. Here, the term BC is known as redundant term. In this way we use this theorem to simply the Boolean algebra. Conditions for applying Redundancy theorem are: 1. Three variables must present in the expression. Here A, B and C are used as variables. 2. Each variables is repeated twice. 3. One variable must present in complemented form. After applying this theorem we can only take those terms which contains the complemented variable. Proof – We can also prove it like this: Y = AB + A'C + BC Y = AB + A'C + BC.1 Y = AB + A'C + BC. (A + A') Y = AB + A'C + ABC + A'BC Y = AB (1 + C) + A’C (1 + B) Y = AB + A'C Example-1. F = AB + BC' + AC Here, we have three variables A, B and C and all are repeated twice. The variable C is present in complemented form. So, all the conditions are satisfied for applying this theorem. After applying Redundancy theorem we can write only the terms containing complemented variables (i.e. C) and omit the Redundancy term i.e., AB. .'. F = BC' + AC Example-2. F = (A + B). (A' + C). (B + C) Three variables are present and all are repeated twice. The variable A is present in complemented form. Thus, all the three conditions of this theorem is satisfied. After applying Redundancy theorem we can write only the terms containing complemented variables (i.e., A) and omit the Redundancy term i.e., (B + C). .'. F = (A + B). (A' + C) Consider the following equation: Y = AB + A'C + BC The third product term BC is a redundant consensus term. If A switches from 1 to 0 while B=1 and C=1, Y remains 1. During the transition of signal A in logic gates, both the first and second term may be 0 momentarily. The third term prevents a glitch since its value of 1 in this case is not affected by the transition of signal A. Thus it is important to remove Logic Redundancy because it causes unnecessary network complexity and raises the cost of implementation.

Some theorems on Nested Quantifiers Quantifiers are expressions that indicate the scope of the term to which they are attached, here predicates. A predicate is a property the subject of the statement can have. For example, in the statement “the sum of x and y is greater than 5”, the predicate ‘Q’ is- sum is greater than 5, and the statement can be represented as Q(x, y) where x and y are variables. The scope of a quantifier or a quantification is the range in the formula that the quantifier engages in. Types of quantification or scopes: 1. Universal(∀) – The predicate is true for all values of x in the domain. 2. Existential(∃) – The predicate is true for at least one x in the domain. To know the scope of a quantifier in a formula, just make use of Parse trees. Two quantifiers are nested if one is within the scope of the other. • Example-1: ∀x ∃y (x+y=5) Here ‘∃’ (read as-there exists) and ‘∀’ (read as-for all) are quantifiers for variables x and y. The statement can be represented as- ∀x Q(x) Q(x) is ∃y P(x, y) Q(x)-the predicate is a function of only x because the quantifier applies only to variable x. P(x, y) is (x + y = 5) • Example-2 ∀x ∀y ((x> 0)∧(y< 0) → (xy< 0)) (in English) For every real number x and y, if x is positive and y is negative, implies xy is negative. again, ∀x Q(x) where Q(x) is ∀y P(x, y) Example to convert a statement into a nested quantifiers formula: “There is a pupil in this lecture who has taken at least one course in Discrete Maths.” A statement consists of quantifiers and predicates, split it into it's two constituents. Here x and y are the pupil and the course and their respective quantifiers are attached in front of them. Write it down as- For some x pupil, there exist a course in Discrete Maths such that x has taken y. ∃x ∃y P (x, y), where P (x, y) is "x has taken y". • Theorem-1: The order of nested existential quantifiers can be changed without changing the meaning of the statement. • • Theorem-2: The order of nested universal quantifiers can be changed without changing the meaning of the statement. • • Example-3: Assume P(x, y) is xy=8, ∃x ∃y P(x, y) domain: integers Translates to- There is an integer x for which there is an integer y such that xy = 8, which is same as- There is a pair of integers x, y for which xy = 8. Meaning ∃x ∃y P(x, y) is equivalent to ∃y ∃x P(x, y). Similarly, Assume P(x, y) is (xy = yx). ∀x ∀y P(x, y) domain: real numbers Translates to- For all real numbers x, for all real numbers y, xy = yx or, For every pair of real numbers x, y, xy = yx. Again ∀x ∀y P(x, y) is equivalent to ∀y ∀x P(x, y). However, when the nested quantifiers are not same, changing the order changes meaning of statement. • Example-4: Assume P(x, y, z) is (x + y = z). ∀x ∀y ∃z P(x, y, z) domain: real numbers Translates to- For all real numbers x and y there is a real number z such that x + y = z (True) ∀z ∃x ∃y P(x, y, z) domain: real numbers There is a real number z such that for all real numbers x and y, x + y = z (False) Negation of nested quantifiers: • Theorem-3 To negate a sequence of nested quantifiers, you change each quantifier in the sequence to the other type and then negate the predicate. So the negation of ∀x ∃y : P(x, y) is ∃x ∀y : ~P(x, y) • Example-5: “ ∃x at Cornell, x is at least 18 years old.” To disagree with this, you’re negating the statement by flipping the ∃ to ∀ and then negating the predicate: “ ∀x at Cornell such that x is not at least 18 years old.”

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Mathematics | Introduction to Propositional Logic What is Logic? Logic is the basis of all mathematical reasoning, and of all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements Which in Simple English means “There exists an integer that is not the sum of two squares”. Importance of Mathematical Logic The rules of logic give precise meaning to mathematical statements. These rules are used to distinguish between valid and invalid mathematical arguments. Apart from its importance in understanding mathematical reasoning, logic has numerous applications in Computer Science, varying from design of digital circuits, to the construction of computer programs and verification of correctness of programs. Propositional Logic What is a proposition? A proposition is the basic building block of logic. It is defined as a declarative sentence that is either true or false, but not both. The Truth Value of a proposition is True (denoted as T) if it is a true statement, and False (denoted as F) if it is a false statement. Some sentences that do not have a truth value or may have more than one truth value are not propositions. To represent propositions, propositional variables are used. By Convention, these variables are represented by small alphabets such as . The area of logic which deals with propositions is called propositional calculus or propositional logic. It also includes producing new propositions using existing ones. Propositions constructed using one or more propositions are called compound propositions. The propositions are combined together using Logical Connectives or Logical Operators. Truth Table Since we need to know the truth value of a proposition in all possible scenarios, we consider all the possible combinations of the propositions which are joined together by Logical Connectives to form the given compound proposition. This compilation of all possible scenarios in a tabular format is called a truth table. Most Common Logical Connectives- 1. Negation 2. Conjunction 3. Dis-junction 4. Exclusive Or 5. Implication 6. Bi-conditional or Double Implication

Mathematics | Lagrange’s Mean Value Theorem Difficulty Level : Easy Last Updated : 16 Jul, 2021 Suppose f:[a,b]\rightarrow R be a function satisfying these conditions: 1) f(x) is continuous in the closed interval a ≤ x ≤ b 2) f(x) is differentiable in the open interval a < x < b Then according to Lagrange’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that: f'(c)=\frac{f(b)-f(a)}{b-a} In simple words, Lagrange’s theorem says that if there is a path between two points A(a, f(a)) and B(b, f(a)) in a 2-D plain then there will be at least one point ‘c’ on the path such that the slope of the tangent at point ‘c’, i.e., (f ‘ (c)) is equal to the average slope of the path, i.e., f'(c)=\frac{f(b)-f(a)}{b-a} Example: Verify mean value theorem for f(x) = x2 in interval [2,4]. Solution: First check if the function is continuous in the given closed interval, the answer is Yes. Then check for differentiability in the open interval (2,4), Yes it is differentiable. {f}'(x)=2x f(2) = 4 and f(4) = 16 \frac{f(b)-f(a)}{b-a} = \frac{16-4}{4-2}=6 Mean value theorem states that there is a point c ∈ (2, 4) such that {f}'(c)=6 But {f}'(x)=2x which implies c = 3. Thus at c = 3 ∈ (2, 4), we have {f}'(c)= 6

Notes on Chemical Bonding Chemical bond:- Chemical bond is the attractive force which holds various constituents together in a molecule. There are three types of chemical bonds: Ionic Bond, Covalent Bond, Co-ordinate Bond. Octet Rule: Atoms form chemical bonds in order to complete their octet i.e. eight electrons in their valence shell. Lewis Structures: Pair of bonded electrons is by means of a ‘dash’ (-) usually called a ‘bond’. Lone pairs or ‘non-bonded’ electrons are represented by ‘dots’. Electrons present in the last shell of atoms are called valence electrons. Exceptions to the Octet Rule: Species with odd number of electrons: NO, NO2, Incomplete octet for the central atom: LiCl, BeH2 and BCl3 Expanded octet for the central atom: PF5, SF6 and H2SO4 Formal Charge: Formal charge is the difference between the number of valence electrons in an isolated atom and number of electrons assigned to that atoms in Lewis structure. Formal charge = [Total number of valence electrons in the free atom ) - (Total number of lone pairs of electrons) -1/2(Total number of shared electrons i.e. bonding electrons)] Resonance: For molecules and ions showing resonance it is not possible to draw a single Lewis structure. All the properties of such species can only be explained by two or more Lewis structures. Example: Resonance of O3 Ionic Bonding: Formation of Ionic Bond: Formation of ionic bond takes place between a metal and a non-metal by transfer of electron. Formation of gaseous cations A(g) + I.E. → A+ (g) + e Ionization Energy Formation of gaseous anions X(g) + e → X- (g) + E.A Electron Affinity Packing of ions of opposite charges to form ionic solids A+ (g) + X- (g) →AX (s) +Energy Lattice energy Conditions required of formation of ionic bonds: Low I.E of cation. High E.A of anion. High lattice energy. Covalent Bonding: Covalent bond is formed between two non-metals by sharing of electrons. Electron pairs which participate in bonding are called bond pairs. Electron pairs which do not participate in bonding are called lone pairs. There could be single, double or triple covalent bonds between two elements depending on the number of electrons being shared. VSEPR (Valence Shell Electron Pair Repulsion) Theory: The shape of the molecule is determined by repulsions between all of the electron pairs present in the valence shell. Order of the repulsion: Lone pair↔️ Lone pair > Lone pair↔️ Bond pair > Bond pair↔️ Bond pair. Repulsion among the bond pairs is directly proportional to the bond order and electronegativity difference between the central atom and the other atoms.

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💎Notes on Gravitation and Projectile Gravitation:- Kepler’s first law (law of elliptical orbit):- A planet moves round the sun in an elliptical orbit with sun situated at one of its foci. Kepler’s second law (law of areal velocities):- A planet moves round the sun in such a way that its areal velocity is constant. Kepler’s third law (law of time period):- A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.Keplers Law of Planetary Motion T2 ∝ R3 Here R is the radius of orbit. T2 = (4π2/GM)R 3 Newton’s law of gravitation:- Every particle of matter in this universe attracts every other particle with a forcer which varies directly as the product of masses of two particles and inversely as the square of the distance between them. F= GMm/r2 Here, G is universal gravitational constant. G = 6.67 ´10 -11 Nm2 / kg2 Dimensional formula of G: G = Fr2/Mm =[MLT-2][L2]/[M2] = [M-1L3T-2] Acceleration due to gravity (g):- g = GM/R2 Variation of g with altitude:- g' = g(1- 2h/R), if h<<R. Here R is the radius of earth and h is the height of the body above the surface of earth. Variation of g with depth:- g' = g(1- d/R). Here g' be the value of acceleration due to gravity at the depth d. Variation with latitude:- At poles:- θ = 90°, g' = g At equator:- θ = 0°, g' = g (1-ω2R/g) Here ω is the angular velocity. As g = GMe/Re2 , therefore gpole > gequator Gravitational Mass:- m = FR2/GM Gravitational field intensity:- E = F/m = GM/r2 Weight:- W= mg Gravitational intensity on the surface of earth (Es):- Es = 4/3 (πRρG) Here R is the radius of earth, ρ is the density of earth and G is the gravitational constant. Gravitational potential energy (U):- U = -GMm/r (a) Two particles: U = -Gm1m2/r (b) hree particles: U = -Gm1m2/r12 – Gm1m3/r13 – Gm2m3/r23 Gravitational potential (V):- V(r) = -GM/r At surface of earth, Vs= -GM/R Here R is the radius of earth. Escape velocity (ve):- It is defined as the least velocity with which a body must be projected vertically upward in order that it may just escape the gravitational pull of earth. ve = √2GM/R or, ve = √2gR = √gD Here R is the radius of earth and D is the diameter of the earth. Escape velocity (ve) in terms of earth’s density:- ve = R√8πGρ/3 Orbital velocity (v0):- v0 = √GM/r If a satellite of mass m revolves in a circular orbit around the earth of radius R and h be the height of the satellite above the surface of the earth, then, r = R+h So, v0 = √MG/R+h = R√g/R+h In the case of satellite, orbiting very close to the surface of earth, then orbital velocity will be, v0 = √gR Relation between escape velocity ve and orbital velocity v0 :- v0= ve/√2 (if h<<R) Time period of Satellite:- Time period of a satellite is the time taken by the satellite to complete one revolution around the earth. T = 2π√(R+h)3/GM = (2π/R)√(R+h)3/g If h<<R, T = 2π√R/g Height of satellite:- h = [gR2T2/4π2]1/3 – R Energy of satellite:- Kinetic energy, K = ½ mv02 = ½ (GMm/r) Potential energy, U = - GMm/r Total energy, E = K+U = ½ (GMm/r) + (- GMm/r) = -½ (GMm/r) Gravitational force in terms of potential energy:- F = – (dU/dR) Acceleration on moon:- gm = GMm/Rm2 = 1/6 gearth Here Mm is the mass of moon and Rm is the radius of moon. Projectile:- Projectile fired at angle α with the horizontal:- If a particle having initial speed u is projected at an angle α (angle of projection) with x-axis, then, Time of Ascent, t = (u sinα)/g Total time of Flight, T = (2u sinα)/g Horizontal Range, R = u2sin2α/g Maximum Height, H = u2sin2α/2g Equation of trajectory, y = xtanα-(gx2/2u2cos2α) Instantaneous velocity, V=√(u2+g2t2-2ugt sinα) and β = tan-1(usinα-gt/ucosα) Projectile fired horizontally from a certain height:- Equation of trajectory: x2 = (2u2/g)y Time of descent (timer taken by the projectile to come down to the surface of earth), T = √2h/g Horizontal Range, H = u√2h/g. Here u is the initial velocity of the body in horizontal direction.

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♦️Revision Notes on Arithmetic Progression♦️ If ‘a’ is the first term and ‘d’ is the common difference of the arithmetic progression, then its nth term is given by an = a+(n-1)d The sum, Sn of the first ‘n’ terms of the A.P. is given by Sn = n/2 [2a + (n-1)d] If Sn is the sum of n terms of an A.P. whose first term is ‘a’ and last term is ‘l’,Sn = (n/2)(a + l) If common difference is d, number of terms n and the last term l, then Sn = (n/2)[2l-(n -1)d] If a fixed number is added or subtracted from each term of an A.P., then the resulting sequence is also an A.P. and it has the same common difference as that of the original A.P. If each term of A.P is multiplied by some constant or divided by a non-zero fixed constant, the resulting sequence is an A.P. again. If a1, a2, a3, …, an andb1, b2, b3, …, bn, are in A.P. then a1+b1, a2+b2, a3+b3, ……, an+bn and a1–b1, a2–b2, a3–b3, ……, an–bn will also be in A.P. Suppose a1, a2, a3, ……,an are in A.P. then an, an–1, ……, a3, a2, a1 will also be in A.P. If nth term of a series is tn = An + B, then the series is in A.P. If a1, a2, a3, ……, an are in A.P., then a1 + an = a2 + an–1 = a3 + an–2 = …… and so on. In order to assume three terms in A.P. whose sum is given, they should be assumed as a-d, a, a+d. Four terms of the A.P. whose sum is given should be assumed as a-3d, a-d, a+d, a+3d Five convenient numbers in A.P. a–2b, a–b, a, a+b, a+2 b. In general, we take a – rd, a – (r – 1)d, …., a – d, a, a + rd in case we have to take (2r + 1) terms in an A.P. Likewise, any 2r terms of an A.P. should be assumed as: a – (2r-1)d, a – (2r – 3)d, …., a – d, a, a + d, ………….. , a+(2r-3)d, a + (2r-1)d. The arithmetic mean of two numbers ‘a’ and ‘b’ is (a+b)/2. The terms A1, A2, ….. , An are said to be arithmetic means between a and b if a, A1, A2, ….. , An, bis an A.P. Clearly, ‘a’ is the first term, ‘b’ is the (n+2)th term and ‘d’ is the common difference. Then, we have b = a+(n+2-1)d = a+(n+1)d Hence, this gives ‘d’ = (b-a)/(n+1) ➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖ Algebra Revision Notes on Arithmetic Progression