Mathematıcal Economıcs (ENG) — Ünite 1 Soru-Cevap

A set is a collection of elements, items, and objects, where the collection is well-defined, meaning the elements in the set share a common trait. In other words, whether an element belongs to the set must be clear-cut.

A subset is a set whose elements are part of another, generally, larger set.

The Cartesian product is the set of all possible ordered combinations where each element of the combination comes from a different set that enters the operation. The Cartesian product of two non-empty sets A and B is A × B = {(a, b) | a ∈ A and b ∈ B}.

The union of sets contains all the elements of the relevant sets and the intersection of sets contains the elements common to the relevant sets. Let A and B be two non-empty sets, the union of set A and set B is A ∪ B = {x| x ∈ A or x ∈ B} where x represents elements in both sets A and B.

The intersection of A and B is A ∩ B = {x| x ∈ A and x ∈ B} where x is the common element of both sets A and B.

The difference between two sets can be thought of as the elimination of elements from one set that is similar to the other. The difference between the sets A and B, in this order, is the set of elements in A that do not belong to B. Symbolically the difference between A and B, in this order, can be represented with either A – B or A \ B.

A function, mapping or transformation is a relation governed by a rule where the second element of the relation is matched to a single element in the first set of the relation. In other words, the function matches each element in the range set, which is represented by the second element in an ordered pair of a relation, to a single element of the domain, which is the set of input elements.

The linear function, in a univariate context, is a function whose graph is a straight line and has the
form f (x) = ax + b where a and b are real numbers and a≠0. In a multivariate context, it is a function that satisfies the linearity conditions.

A function written in the form y = f (x) = bx where x is any real number b>0, b≠1, is called an exponential function with base.

Simple interest is calculated on the principle; it is a fixed percentage of the initial amount.

Compound interest accrues on both the principal and the accumulated interest of previous periods.

Functions that are inverse of exponential functions are called logarithmic functions. As a general
expression, if an exponential function is expressed as y = a^x, then the logarithmic function of this
expression is formulated as log_a y = x with the base of a where a>0 and a≠1. Alternatively, we can also write x = log_a y + y = a^x. The domain of the function is the positive real numbers, and the image set is the real numbers. 

A power function is of the form y = cxⁿ where c and n are any real numbers and c is known as the coefficient.

In simple terms, a continuous function is a graph that can be drawn from one end of the coordinate
system to the other without lifting the pen.

If it is necessary to lift the pen to graph a function, then it is not a continuous but a discrete function. In other words, the function of the graph which is not connected with each other is a discrete or discontinuous function.

A jump discontinuity from the right x = a occurs if the right hand of the function exists but is not
equal to f (a), saying, the left and right-handed limits aren’t equal. That is, when drawing the graph of the function, the pen is lifted once from the paper but from a horizontal point of view, there seems to be a complete line.

A removable discontinuity occurs when there is a rational expression with common factors in the
numerator and denominator. Since these factors can be canceled, the discontinuity is “removable.”

An infinite discontinuity occurs when the curve has a vertical asymptote. This is also called discontinuity. When the graph of a function f (x) has the line x = k, as a vertical asymptote, then f (x) becomes positively or negatively infinite as x " k⁺ or x " k^–. Then, function f (x) is said to have an infinite discontinuity.

The function can be said to be convex if on the graph of the function the line goes above, or on the graph. A convex function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.

The function can be said to be concave if on the graph of the function the line goes below, or on the graph. A concave function is a continuous function whose value at the midpoint of every interval in its domain does not exceed the arithmetic mean of its values at the ends of the interval.