Rigorous and Logical Arguments

Mathematical proofs are rigorous and logical arguments that validate the truth of mathematical statements

Transforming Conjectures into Theorems

The purpose of mathematical proofs is to transform conjectures into universally accepted theorems

Built upon Axioms

Mathematical proofs are built upon axioms, which are fundamental truths accepted without proof

Deductive Proof

Deductive proof starts from axioms and established theorems and proceeds logically to the conclusion

Proof by Counterexample

Proof by counterexample refutes a statement by providing a single instance where it fails

Proof by Exhaustion

Proof by exhaustion verifies a statement by considering all possible cases

Beginning with Known Truths

Deductive reasoning in mathematical proofs begins with known truths and applies logical steps to reach a new conclusion

Precise Logical Reasoning

Deductive reasoning in mathematical proofs requires precise logical reasoning and careful consideration of all conditions

Ensuring Validity of Conclusion

Deductive reasoning in mathematical proofs ensures that the conclusion is valid by considering all possible scenarios

Demonstrating Equivalence of Expressions

Proving mathematical identities involves demonstrating that two expressions are equivalent under all circumstances

Expanding and Simplifying Expressions

Algebraic identities are often proved by expanding and simplifying expressions to show their equivalence

Using Fundamental Relationships and Properties

Trigonometric identities are often proved by employing fundamental relationships and properties derived from the Pythagorean theorem