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.