Mathematical Reasoning and Proof: Concepts and Problem Solutions

Vocabulary practice flashcards covering fundamental concepts of mathematical reasoning, definitions, conditional propositions, and logical transformations from the Tartu Ülikool course module.

Content of a Concept (Intension)

The set of essential defining characteristics and properties of a concept; for instance, the content of the concept 'circle' includes being a round planar geometric figure that has no corners.

Scope of a Concept (Extension)

The collection of all individual objects to which a concept applies; for instance, the scope of the concept 'circle' consists of all possible circles of varying sizes.

Universal Triangle Properties

Geometric properties true for all triangles, including having three angles, an interior angle sum of $180^\circ$, medians intersecting at one point, at least one acute angle, all three vertices lying on a circle, and the sum of the two shorter sides being greater than the longest side.

Overly Broad Definition

A logical error in definition occurring when the defining phrase includes objects outside the defined class; for example, defining a square merely as 'a rectangle', which fails because there exist rectangles that are not squares.

Overly Narrow Definition

A logical error in definition occurring when the definition unnecessarily excludes valid instances of the term; for example, defining a triangle as 'a polygon with three equal sides', which only defines an equilateral triangle.

Altitude of a Triangle

A perpendicular line segment drawn from a vertex of a triangle to the line containing the opposite side (or to the opposite side or its extension).

Hypothesis (Premise)

The given antecedent or condition in a conditional 'if…, then…' statement; for example, $2x - 1 = 5$ in the statement 'If $2x - 1 = 5$, then $x = 3$'.

Conclusion (Assertion)

The consequent or claim that follows from the hypothesis in an 'if…, then…' statement; for example, $x = 3$ in the statement 'If $2x - 1 = 5$, then $x = 3$'.

Converse Statement

A statement formed by interchanging the hypothesis and conclusion of an original implication; for example, the converse of 'If $x = -5$, then $x^2 = 25$' is 'If $x^2 = 25$, then $x = -5$' (which is false because $x = 5$ is also a valid solution).

Contrapositive Statement

A statement formed by swapping and negating both the hypothesis and the conclusion of an original implication; for example, 'If $x^2 \neq 25$, then $x \neq -5$', which remains true whenever 'If $x = -5$, then $x^2 = 25$' is true.