Overview
This lecture covers proof by contradiction, a fundamental technique in algebraic methods, including key steps, example proofs, and important definitions related to rationality and irrationality.
Proof by Contradiction: Method Overview
- Start by assuming the opposite of the statement you want to prove.
- Perform mathematical operations to develop the assumption.
- Identify a logical or mathematical contradiction.
- Conclude the original statement must be true, clearly stating the contradiction.
Worked Examples
- To prove no integers ( x ) and ( y ) exist such that ( 15x + 20y = 1 ), assume such integers exist and show resulting equations cannot yield integer solutions.
- To prove no positive integers ( a ) (with ( a ) odd) and ( b ) exist such that ( a + 4b = 4\sqrt{ab} ), assume the existence and manipulate algebraically to show ( a ) must be even, contradicting ( a ) being odd.
Proofs Involving Rationality and Irrationality
- A rational number can be written as ( A/B ) where ( A ) and ( B ) are integers with no common factors.
- Proof by contradiction can demonstrate statements about rational or irrational results, such as:
- If ( x ) and ( y ) are rational, assuming ( x + y ) is irrational leads to a contradiction if written as a single fraction.
- To prove ( \sqrt{2} ) is irrational, assume ( \sqrt{2} = A/B ), show both ( A ) and ( B ) become even, contradicting no common factors.
- To prove ( \log_2 7 ) is irrational, assume rationality and derive ( 2^a = 7^b ), which cannot hold for nonzero integers due to mismatched prime factors.
Key Terms & Definitions
- Proof by Contradiction — Assuming the negation of a statement, deriving a contradiction, and concluding the original statement must be true.
- Rational Number — A number expressible as ( A/B ) where ( A ) and ( B ) are integers, ( B \neq 0 ), with no common factors.
- Irrational Number — A real number that cannot be expressed as a ratio of two integers.
Action Items / Next Steps
- Practice more proof by contradiction exercises.
- Review definitions and properties of rational and irrational numbers.
- Ensure to always include a clear conclusion in contradiction proofs.