Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) key terms

A declarative sentence that is either definitely true or definitely false, but not both at once. Questions, commands and exclamations are not statements.

The label T (true) or F (false) assigned to a statement. Compound statements take truth values determined by their components and the connectives joining them.

A table listing every possible combination of truth values of the component statements together with the resulting truth value of the compound statement. A table with n components has 2^n rows.

A compound statement whose truth value is T in every row of its truth table. Its negation is a contradiction, which is false in every row.

Two compound statements are logically equivalent when their truth tables match row for row. This is written with the double arrow and allows one form to replace the other in arguments.

For the conditional p implies q, the contrapositive is not q implies not p. It is logically equivalent to the original conditional, unlike the converse and the inverse.

The statement obtained by swapping the hypothesis and conclusion of a conditional: from p implies q we get q implies p. The converse is not generally equivalent to the original.

The words 'for all' and 'there exists' that fix how many objects a statement covers. Their negations swap: not-for-all becomes there-exists-not, and not-there-exists becomes for-all-not.

A set containing exactly one element. Its power set therefore has 2^1 = 2 subsets, namely the empty set and the set itself.

The collection of all subsets of a given set, including the empty set and the set itself. A set with n elements has a power set with 2^n elements.

Set A is a subset of set B when every element of A also belongs to B. Every set is a subset of itself, and the empty set is a subset of every set.

The set of all elements of the universal set that are not in the given set. Complementation reverses inclusion: if A is inside B, the complement of B is inside the complement of A.

Rules for negating unions and intersections: the complement of A union B equals complement A intersect complement B, and the complement of A intersect B equals complement A union complement B.

The size of a matrix stated as rows by columns, for example 2 x 3 means two rows and three columns. Two matrices can be added only when their orders are identical.

The matrix formed by writing rows as columns and columns as rows, denoted with a superscript T. Transposing twice returns the original matrix, and the transpose of a product reverses the order.

A single number computed from a square matrix that tests invertibility and appears in Cramer's rule. For a 2 x 2 matrix with entries a, b in the first row and c, d in the second, it equals ad minus bc.

A square matrix whose determinant equals zero, so it has no multiplicative inverse. Its rows or columns are linearly dependent, and the associated system may have no solution or infinitely many.

For a square matrix A with non-zero determinant, the matrix that multiplies with A to give the identity matrix. For 2 x 2 it is the adjoint divided by the determinant.

A method of solving a square system of linear equations in which each unknown equals a ratio of two determinants, the denominator being the coefficient determinant and the numerator replacing one column by the constants.

The instantaneous rate of change of a function with respect to its variable, obtained as the limit of the difference quotient. It gives the slope of the tangent line at a point.

A rule for differentiating a composite function: differentiate the outer function at the inner function and multiply by the derivative of the inner function. It extends to three or more nested functions.

The rule that the derivative of a product of two functions is the first times the derivative of the second plus the second times the derivative of the first.

For a quotient of two functions, the derivative is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

The derivative of the total cost function with respect to quantity, approximating the extra cost of producing one more unit. It is also called the rate of change of total cost.

The derivative of total revenue with respect to quantity, approximating the change in revenue from selling one additional unit. Profit is maximised where marginal revenue equals marginal cost.

Points where a differentiable function turns, found by setting the first derivative to zero. The second derivative decides the type: negative gives a maximum, positive gives a minimum.

The proportional change in quantity demanded divided by the proportional change in price, often written as the price divided by quantity times the derivative of quantity with respect to price. Demand is elastic when its magnitude exceeds one.

The family of all antiderivatives of a function, written with an arbitrary constant of integration. Differentiating the result returns the original integrand.

The limit of a sum of areas over an interval, evaluated using the fundamental theorem as the antiderivative at the upper limit minus that at the lower limit. It produces a number, not a family of functions.

A technique based on the product rule, written as the integral of u dv equals uv minus the integral of v du. It suits products of algebraic and logarithmic or exponential factors.

The region between a curve and the x-axis over an interval, computed as the definite integral of the function. Regions below the axis contribute negative values unless absolute value is used.

The gain to buyers measured as the area between the demand curve and the market price line, up to the quantity traded. It is found by integrating the difference between demand price and market price.

The gain to sellers measured as the area between the market price line and the supply curve, up to the quantity traded. Integrating market price minus supply price over that quantity gives it.

The order is the highest derivative present in the equation; the degree is the power of that highest derivative once the equation is free of radicals and fractions in derivatives.

The solution of a differential equation containing as many arbitrary constants as the order of the equation. It represents a whole family of curves.

A solution obtained from the general solution by fixing the arbitrary constants using given initial conditions. It corresponds to one specific curve satisfying the equation.

The distribution of the number of successes in n independent trials, each with the same success probability p. Its mean is np and its variance is npq, where q equals one minus p.

A distribution for the count of rare events in a fixed interval, defined by a single parameter m, the mean number of occurrences. Its mean and variance both equal m.

A single experiment with exactly two outcomes, usually called success and failure, with constant probabilities. A sequence of independent Bernoulli trials underlies the binomial distribution.

A function giving the probability that a discrete random variable takes each particular value. All its values are non-negative and they sum to one over the possible values.

A real-valued function defined on the sample space that assigns a number to each outcome. It may be discrete, taking isolated values, or continuous, taking any value in an interval.

The probability-weighted average of a random variable, found by summing each value multiplied by its probability. It is the long-run mean and is denoted E of X.

The expected value of the squared deviation from the mean, computed as E of X squared minus the square of E of X. Its positive square root is the standard deviation.

The positive square root of the variance, expressed in the same units as the variable. It measures the spread of values around the mean.

A statistical measure expressing the change in a variable or group of variables relative to a chosen base period, usually written with the base value set to 100.

A weighted price index that uses base-period quantities as weights. It tends to overstate price rises because it ignores substitution away from goods that became dearer.

A weighted price index that uses current-period quantities as weights. It tends to understate price rises relative to the Laspeyres measure because the weights already reflect substitution.

A sequence of values of a variable recorded at successive, usually equally spaced, time points. Its movement is analysed as trend, seasonal, cyclical and irregular components.

The long-term general direction of a time series, obtained by smoothing out short-term fluctuations. It may be rising, falling or stable over the period studied.

A smoothed value obtained by averaging a fixed number of consecutive observations and shifting the window forward one period at a time. It removes short-term fluctuations and reveals trend.