Elementary Mathematics Practice Flashcards

Practice flashcards covering introductory concepts of elementary mathematics, including logic, set theory, functions, number systems, and basic Euclidean geometry based on the student's lecture notes.

Proposition (Sud)

A meaningful declarative sentence that is either true or false.

Negation (Negacija suda A)

A proposition that denies what proposition $A$ claims.

Conjunction (Konjukcija sudova A i B)

A composite proposition that is true if and only if both propositions $A$ and $B$ are true.

Disjunction (Disjunkcija sudova A i B)

A composite proposition that is false if and only if both propositions $A$ and $B$ are false.

Exclusive Disjunction (Ekskluzivna disjunkcija)

A composite proposition that is true if and only if exactly one of the propositions $A$ or $B$ is true.

Implication (Implikacija sudova A i B)

A composite proposition that is false if and only if the antecedent proposition $A$ is true and the consequent proposition $B$ is false.

Equivalence (Ekvivalencija sudova A i B)

A composite proposition that is true if and only if both propositions $A$ and $B$ have the same truth value (both true or both false).

Equality of Composite Propositions

Two propositions are equal when their corresponding semantic (truth) tables match.

Tautology

A composite proposition that is always true regardless of the truth values of the individual propositions that compose it.

Converse (Obrat suda)

For a given implication $A \rightarrow B$, the converse is the proposition $B \rightarrow A$.

Contrapositive (Obrat po kontrapoziciji)

For a given implication $A \rightarrow B$, the contrapositive is the proposition $\neg B \rightarrow \neg A$.

Inverse (Suprotni sud ili inverzija)

For a given implication $A \rightarrow B$, the inverse is the proposition $\neg A \rightarrow \neg B$.

Basic Concept (Osnovni pojam)

A concept that is considered known and is not defined using other concepts, such as point, line, set, space, or plane.

Derived Concept (Izvedeni pojam)

A concept defined or described using basic concepts or previously defined derived concepts.

Definition

The stating of necessary and sufficient features of a concept connected by a logical sentence or symbolic notation.

Axiom (Aksiom)

A basic statement that is taken as true within a theory and is not proven.

Postulate (Postulat)

A starting statement taken without proof that usually expresses a condition a concept or relationship must satisfy.

Theorem (Teorem)

A mathematical statement whose truth is established through a proof, consisting of a premise ($P$) and an assertion ($Q$).

Set (Skup)

A basic mathematical concept that is not formally defined, consisting of elements sharing a common property.

Cardinal Number

The total number of elements contained within a set.

Empty Set (Prazan skup)

A set that contains no elements, denoted by the symbol $\text{\text{\text{\text{\text{\text{\text{\text{Ø}}}}}}}}$.

Universal Set (Univerzalni skup)

A broader set from which elements are taken for a specific context.

Subset (Podskup)

Set $A$ is a subset of set $B$ ($A \text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}}\text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}} B$) if every element of $A$ is also an element of $B$.

Equal Sets

Two sets $A$ and $B$ are equal if every element of $A$ is an element of $B$ and conversely ($A \text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}}\text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}} B$ and $B \text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}}\text{\text{\text{\text{\text{\text{\text{\text{∣}}}}}}}} A$).

Proper Subset (Pravi podskup)

Set $A$ is a proper subset of set $B$ ($A \text{\text{\text{\text{\text{\text{\text{\text{∢}}}}}}}} B$) if every element of $A$ is in $B$ and there exists at least one element in $B$ that is not in $A$.

Power Set (Partitivni skup)

The set of all subsets of a given set $A$, denoted as $P(A)$. If a set has $n$ elements, its power set has $2^n$ subsets.

Union of Sets (Unija skupova)

The set containing all elements that belong to set $A$ or set $B$: $A \text{\text{\text{\text{∪}}}} B = \text{\text{\text{\text{{}}}}x : x \text{\text{\text{\text{∈}}}} A \text{\text{\text{\text{∨}}}} x \text{\text{\text{\text{∈}}}} B\text{\text{\text{\text{}}}}}$.

Intersection of Sets (Presjek skupova)

The set containing all elements that belong to both set $A$ and set $B$: $A \text{\text{\text{\text{∩}}}} B = \text{\text{\text{\text{{}}}}x : x \text{\text{\text{\text{∈}}}} A \text{\text{\text{\text{∧}}}} x \text{\text{\text{\text{∈}}}} B\text{\text{\text{\text{}}}}}$.

Disjoint Sets

Sets $A$ and $B$ for which the intersection is empty: $A \text{\text{\text{\text{∩}}}} B = \text{\text{\text{\text{Ø}}}}$.

Set Difference (Diferencija skupova)

The set of elements that belong to set $A$ but do not belong to set $B$: A \text{\text{\text{\text{∖}}}} B = \text{\text{\text{\text{{}}}}x : x \text{\text{\text{\text{∈}}}} A \text{\text{\text{\text{∧}}}} x \text{\text{\text{\text{∉}}}} B\text{\text{\text{\text{}}}}}.

Complement of a Set (Komplement skupa)

The set of all elements in the universal set $U$ that are not in set $A$: A^c = \text{\text{\text{\text{{}}}}x : x \text{\text{\text{\text{∈}}}} U \text{\text{\text{\text{∧}}}} x \text{\text{\text{\text{∉}}}} A\text{\text{\text{\text{}}}}}.

Cartesian Product (Kartezijev produkt)

The set of all ordered pairs $(x, y)$ such that $x \text{\text{\text{\text{∈}}}} A$ and $y \text{\text{\text{\text{∈}}}} B$. The number of elements is $n(A) \text{\text{\text{\text{×}}}} n(B)$.

Binary Relation (Binarna relacija)

Any non-empty subset of the Cartesian product $X \text{\text{\text{\text{×}}}} Y$.

Equivalence Relation (Relacija ekvivalencije)

A relation on a set $X$ that is reflexive ($a\text{ρ}a$), symmetric ($a\text{ρ}b \text{\text{⇒}} b\text{ρ}a$), and transitive ($a\text{ρ}b \text{\text{∧}} b\text{ρ}c \text{\text{⇒}} a\text{ρ}c$).

Partial Order Relation (Relacija parcijalnog uređaja)

A relation on a set $X$ that is reflexive, antisymmetric ($a\text{ρ}b \text{\text{∧}} b\text{ρ}a \text{\text{⇒}} a = b$), and transitive.

Function (Funkcija)

A mapping from set $D$ (domain) to set $K$ (codomain) that assigns exactly one element in $K$ to each element in $D$.

Injection (Injekcija)

A function where distinct arguments result in distinct values: x_1 \text{\text{\text{≠}}} x_2 \text{\text{\text{⇒}}} f(x_1) \text{\text{\text{≠}}} f(x_2).

Surjection (Surjekcija)

A function where every element of the codomain $K$ is the image of at least one element from the domain $D$ ($I(f) = K$).

Bijection (Bijekcija)

A function that is both injective and surjective (one-to-one correspondence).

Equipotent Sets (Ekvipotentni skupovi)

Two sets are equipotent if there exists at least one bijection between them.

Mathematical Induction (Matematička indukcija)

A proof technique involving three steps: the Base Case ($n=1$), the Inductive Hypothesis ($n=k$), and the Inductive Step ($n=k+1$).

Prime Number (Prost broj)

A natural number greater than $1$ that is divisible only by $1$ and itself.

Composite Number (Složeni broj)

A natural number greater than $1$ that is not prime.

Euclidean Algorithm (Euklidov algoritam)

A theorem about division with remainder used to find the greatest common divisor ($\text{NZD}$) of two natural numbers.

Dense Set (Gust skup)

A set where between any two elements there exists at least one other element from the same set: $r_1 \text{\text{\text{<}}} \frac{r_1+r_2}{2} \text{\text{\text{<}}} r_2$.

Convex Set (Konveksni skup)

A set in a plane is convex if for any two points $A$ and $B$ in the set, the entire segment $AB$ is contained within the set.

Vertical Angles (Vršni kutovi)

Two angles that share a common vertex and whose sides are opposite rays; they are always congruent.

Parallel Postulate (V. Aksiom o paralelama)

Through a point outside a given line, at most one line can be drawn parallel to the given line.

Centroid (Težšte trokuta)

The point where all three medians of a triangle intersect; it is located two-thirds of the way from each vertex to the midpoint of the opposite side.

Euler Line (Eulerov pravac)

A line in a triangle on which the circumcenter ($S$), the centroid ($G$), and the orthocenter ($H$) lie, satisfying the ratio $|HG| = 2 \text{\text{\text{⋅}}} |SG|$.