Mathematics Fundamentals Quiz

The major subdisciplines of modern mathematics mentioned in the text are number theory, algebra, geometry, and analysis.

There is no general consensus among mathematicians about a common definition for their academic discipline.

A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.

<p>Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation.</p> Signup and view all the answers

<p>Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics.</p> Signup and view all the answers

<p>Reflexivity means that every element is related to itself, which is represented by the property $a \sim a$. Symmetry means that if $a \sim b$, then $b \sim a$. Transitivity means that if $a \sim b$ and $b \sim c$, then $a \sim c$.</p> Signup and view all the answers

<p>An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. An example of an equivalence relation in geometry is the equipollence relation between line segments.</p> Signup and view all the answers

<p>The various notations used are $a \sim b$, $a \equiv b$, $a \sim_R b$, $a \equiv_R b$, or $a \mathop{R} b$.</p> Signup and view all the answers

<p>A binary relation $\sim$ on a set $X$ is an equivalence relation if and only if it is reflexive, symmetric, and transitive.</p> Signup and view all the answers

<p>Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.</p> Signup and view all the answers