Mathematical Foundations Vocabulary

a statement accepted as true without proof, serving as a starting point for reasoning

“Euclid's fifth axiom about parallel lines sparked centuries of mathematical inquiry.”

a statement that has been proven true based on axioms and other theorems

“The Pythagorean theorem relates the sides of a right triangle.”

a proven proposition used as a stepping stone to a larger result

“The proof relied on a clever lemma about prime factorization.”

a proposition that follows directly from a proven theorem

“As a corollary, all equilateral triangles are also equiangular.”

a proposition believed to be true but not yet proven

“The Riemann conjecture remains one of mathematics' greatest unsolved problems.”

a logical argument establishing the truth of a statement

“Her elegant proof reduced the problem to a simple contradiction.”

a statement assumed true as a basis for reasoning

“Einstein's postulate that light speed is constant revolutionized physics.”

a proposed explanation or assumption to be tested

“The null hypothesis assumes no significant difference between groups.”

proving a statement by showing its negation leads to an impossibility

“The irrationality of √2 is elegantly shown by proof by contradiction.”

proving a statement for all natural numbers by establishing a base case and inductive step

“Mathematical induction proves the formula holds for every positive integer.”