CS2004 : 3.1 Mathematical Foundation

What is a variable in mathematics?

A variable is a symbol used to represent a mathematical construct such as numbers, sets, lists, vectors, matrices. They are often represented by lower case letters or Greek letters.

What is a set in mathematics?

set is a collection of objects called elements. Sets can be finite or infinite, have no order, and only contain one copy of an item.

What are some well-known sets?

Some well-known sets include R: Real numbers, Z: Integers, N: Natural numbers (integers ≥ 0), and The alphabet.

What is the cardinality of a set?

The cardinality of a set, represented as |A|, is the number of items in the set.

What is the intersection and union of two sets?

The intersection (A ∩ B) of two sets is the set of elements common to both A and B. The union (A ∪ B) of two sets is the set of all elements that are in either set A or set B.

What is an equation in mathematics?

An equation uses mathematical operators to relate one set of variables or numbers to another set of variables or numbers. Examples include y = mx + c, ax2 + bx + c = 0.

What is a function in mathematics?

A function is a relation that uniquely associates members of one set with members of another set.

What is the exponential function?

he exponential function is represented as f(x) = exp(x) = e^x where e is approximately 2.718.

What are subscripts in mathematics?

A subscript is a natural number that indexes a list of variables. For example, if X is the list (or vector) [x1, ..., xn], then to access any element we use the notation xi, where 1≤i ≤n.

What is the summation notation in mathematics

The summation notation is represented by the Greek letter sigma (Σ). It is used to indicate the addition of a series of numbers. For example, the sum of all elements in a list X would be represented as Σxi (i from 1 to n).

What is the product notation in mathematics?

The product notation is represented by the Greek letter pi (Π). It is used to indicate the multiplication of a series of numbers. For example, the product of all elements in a list X would be represented as Πxi (i from 1 to n).

What is the factorial notation in mathematics?

: The factorial of a number n, represented as n!, is the product of all positive integers less than or equal to n.

What is the difference between permutations and combinations?

Permutations consider order as important while combinations do not. In permutations, different orders count as different outcomes, but in combinations, order doesn't matter.

What are logarithms?

A logarithm is the power to which a number is raised to get some other number. For example, log10 100 = 2 because 10^2 = 100. There are different logarithms using different base units. The most common ones are base 10 logarithms and natural logarithms.

What are some important properties of logarithms?

logb(x) = y means x = b^y.

logb(b) = 1 because b^1 = b for all b ≠ 0.

logb(1) = 0 because b^0 = 1 for all b ≠ 0.

logb(xy) = logb(x) + logb(y).

logb(x/y) = logb(x) - logb(y).

logb(x^y) = y * logb(x).

logb(x) = loga(x) / loga(b) for a change of base.

Logarithms are undefined for x ≤ 0.

What does differentiation refer to in mathematics?

Differentiation is a process in calculus that measures the rate at which a quantity changes. It is used to find the derivative of a function.

What is integration in mathematics?

Integration is the reverse process of differentiation. It's a fundamental concept in calculus used to calculate the area under the curve, the length of curves, and the total value when given a rate of change.

What does solving equations involve in mathematics?

Solving equations involves finding the value(s) of the variable(s) that make the equation true. This can involve a variety of methods including factoring, substitution, and using the quadratic formula.

What are mathematical proofs?

Mathematical proofs are logical arguments that establish the truth of a mathematical statement. They use axioms, definitions, and previously proven statements.

What is statistics in mathematics?

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

What is the importance of summary statistics?

Summary statistics provide simple summaries about the sample and the measures. These summaries may include measures of central tendency like mean and median, or measures of variability like variance and standard deviation.