theorem 🔊
Meaning of theorem
A theorem is a statement that has been proven to be true through logical reasoning and mathematical proof, based on previously established statements such as axioms or other theorems.
Key Difference
A theorem is distinct from other similar terms like 'hypothesis' or 'conjecture' because it is already proven, whereas the others are unproven propositions.
Example of theorem
- The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
- Fermat's Last Theorem, which remained unproven for centuries, was finally solved by Andrew Wiles in 1994.
Synonyms
principle 🔊
Meaning of principle
A fundamental truth or proposition that serves as the foundation for a system of belief or behavior or for a chain of reasoning.
Key Difference
A principle is a general truth or guideline, while a theorem is a mathematically proven statement.
Example of principle
- The principle of conservation of energy states that energy cannot be created or destroyed, only transformed.
- Archimedes' principle explains why objects float or sink in fluids.
law 🔊
Meaning of law
A statement of fact, deduced from observation, to the effect that a particular natural or scientific phenomenon always occurs if certain conditions are present.
Key Difference
A law is typically derived from repeated observations, whereas a theorem is derived from logical proof.
Example of law
- Newton's laws of motion form the foundation of classical mechanics.
- The law of supply and demand governs market economies.
lemma 🔊
Meaning of lemma
A subsidiary or intermediate theorem in an argument or proof.
Key Difference
A lemma is a smaller result used to prove a larger theorem, whereas a theorem is the main proven statement.
Example of lemma
- Euclid's lemma is used in the proof of the fundamental theorem of arithmetic.
- Zorn's lemma is essential in many proofs in set theory.
corollary 🔊
Meaning of corollary
A proposition that follows from (and is often appended to) one already proved.
Key Difference
A corollary is a direct consequence of a theorem, not an independent statement.
Example of corollary
- A corollary to the Pythagorean theorem is that the hypotenuse is the longest side of a right triangle.
- The corollary to the fundamental theorem of calculus connects differentiation and integration.
proposition 🔊
Meaning of proposition
A statement or assertion that expresses a judgment or opinion.
Key Difference
A proposition is a general statement that may or may not be proven, while a theorem is definitively proven.
Example of proposition
- The proposition that all men are created equal is a foundational idea in democratic societies.
- In geometry, a proposition may be a problem to solve or a theorem to prove.
axiom 🔊
Meaning of axiom
A statement or proposition which is regarded as being established, accepted, or self-evidently true.
Key Difference
An axiom is an accepted truth without proof, while a theorem requires proof.
Example of axiom
- Euclid's axioms are the basis for classical geometry.
- The axiom of choice is a fundamental principle in set theory.
conjecture 🔊
Meaning of conjecture
An opinion or conclusion formed on the basis of incomplete information.
Key Difference
A conjecture is an unproven hypothesis, whereas a theorem is proven.
Example of conjecture
- Goldbach's conjecture states that every even integer greater than 2 can be expressed as the sum of two primes.
- The twin prime conjecture remains one of the most famous unsolved problems in mathematics.
hypothesis 🔊
Meaning of hypothesis
A supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation.
Key Difference
A hypothesis is an unproven assumption, while a theorem is a proven statement.
Example of hypothesis
- The Riemann hypothesis is one of the most important unsolved problems in mathematics.
- Einstein's hypothesis about the photoelectric effect led to the development of quantum theory.
formula 🔊
Meaning of formula
A mathematical relationship or rule expressed in symbols.
Key Difference
A formula is an equation or expression, while a theorem is a proven statement or rule.
Example of formula
- The quadratic formula provides solutions to quadratic equations.
- Euler's formula connects complex exponentials with trigonometric functions.
Conclusion
- A theorem is a cornerstone of mathematical reasoning, providing proven truths that form the basis for further exploration.
- Principles can be used when referring to general truths that guide reasoning or behavior, not necessarily mathematical proofs.
- Laws are best when describing universally observed phenomena in science or nature, rather than abstract mathematical truths.
- Lemmas should be used when referring to intermediate results that help in proving larger theorems.
- Corollaries are perfect when discussing direct consequences of a theorem, adding depth to the original statement.
- Propositions are useful for general statements that may or may not be proven, offering flexibility in argumentation.
- Axioms are foundational truths that require no proof, serving as the starting points for logical systems.
- Conjectures are ideal for discussing unproven ideas that inspire further research and exploration.
- Hypotheses are best suited for initial assumptions in scientific or mathematical investigations.
- Formulas are essential when expressing mathematical relationships in a concise symbolic form.