Proof

Introduction to Proof

  • Proof is a logical argument that conclusively demonstrates the truth of a statement.
  • In mathematics, it’s used to unequivocally confirm that a statement is true for all cases, not just some examples.

Direct Proof

  • A direct proof is a type of proof where the conclusion is stated directly.
  • In a direct proof, the argument moves forward in a clear, straightforward line.
  • Direct proofs often take the form “If A, then B” or “B because A”.

Proof by Contradiction

  • Proof by contradiction, also known as reductio ad absurdum, is where you assume the opposite of what you’re trying to prove, then show that this leads to an absurd situation.
  • This type of proof can be very powerful because it contradicts the initial false assumption.
  • Often used when the direct proof seems too complex or isn’t feasible.

Mathematical Induction

  • Mathematical induction is a method of proof used to establish a claim about all natural numbers.
  • Contains two parts: base case and induction step. The base case shows the statement holds for a particular number, often 1. The induction step shows that if the statement holds for some number ‘n’, then it holds for ‘n+1’.
  • This can also be used to prove claims about objects other than natural numbers, like geometric figures or sentences in formal languages.

Set Theory and Proof

  • Set theory is often used in proofs, particularly proofs about numbers.
  • Sets can prove mathematical propositions that are true for a range of numbers.
  • Venn diagrams and Euler diagrams are visual tools that can be used in set theory proofs.

Spotting Errors in Proof

  • As well as constructing proofs, you need to be able to critically examine proofs for errors, either in structure or logical reasoning.
  • Common errors include incorrect assumptions, misapplication of a mathematical theorem or law, and false inferences.
  • Being able to spot these errors is an important skill in mathematical proof.

Using Proof in Real World Applications

  • Mathematical proof isn’t just about abstract concepts – it can also relate to real world applications.
  • For instance, it can be used to prove the validity of algorithm, or that certain variables in physics or engineering formulas are indeed true.
  • Being able to apply proof to real world problems is an important skill for future mathematicians and engineers.