Summary
Highlights
The first step in proving a theorem is to identify its logical structure, most commonly a conditional statement "P implies Q" (if assumed P, then concluded Q). This structure helps in forming a strategy for the proof. Other structures include biconditional statements (P if and only if Q), existential statements (there exists some value), or simply statements of truth.
Various methods can be used to prove conditional statements. A direct proof assumes P and directly derives Q. Proof by contrapositive assumes 'not Q' and derives 'not P'. Proof by contradiction assumes P and 'not Q' and shows that this leads to a logical inconsistency. Lastly, a counterexample is used to disprove a conditional statement by finding one instance where P is true but Q is false. It's crucial not to confuse a single example for a general proof.
A crucial step is to write down the precise mathematical definitions of every word used in the assumptions and conclusions. This helps in understanding what you're working with and what you need to achieve. By clearly defining both the starting point (defined assumptions) and the end point (defined conclusions), the 'gap' for manipulations becomes clearer and often smaller.
Using the example "If X is an even number, then X squared is an even number," the speaker demonstrates defining terms. 'X is even' becomes X = 2P (where P is an integer), and 'X squared is even' becomes X^2 = 2Q (where Q is an integer). The manipulation then involves squaring X (2P)^2 = 4P^2 = 2(2P^2), thus showing X^2 is indeed 2Q where Q = 2P^2, completing the proof.
Beyond the formal structure, truly understanding the statement's meaning, both mathematically and intuitively, is vital. Drawing pictures or geometric representations can often provide significant insight into the relationships between assumptions and conclusions, especially in fields like linear algebra.
While not a proof, working through specific concrete examples can build intuition and help generalize the proof. Additionally, identifying and leveraging relevant theorems from your course material or textbook is often key, as many proofs involve connecting definitions through a few core theorems. Reading proofs of these relevant theorems can also inspire manipulation techniques.
Proving theorems is an iterative process. Don't be discouraged if the first attempt doesn't work. Continually play around with ideas, try new examples, reconsider different proof methods (a direct proof might be difficult, but a contrapositive or contradiction might be easier), and seek new insights. This persistent exploration is often what leads to a breakthrough in proving a theorem.