one of my exam questions was "find the negation of the statement: 'mathwonk is both stupid and lazy'".
YUou cannot do this if you do not know how to form the negation of a statement.
This is because proving a given statement is true, is often done by proving its negation cannot be false. You must also know how to phrase a statement that is the "opposite" of the given statement, namely the "negation" of a given statement. To repeat, before you can prove a given statement is true, you must know exactly what it means to say it is true, and in particular you must know what other statements are equivalent to that statement. This is the game you must learn to play, which requires first learning very well just what it actually means to say a given statement is tue, and what statements are equivalent to each other in terms of truth value. Equivalently, if A is true, then B must be false. Similarly, to prove that A and B cannot both be true is the same as proving that at least one is false. Equivalently, if A is false, then B must be true. Similarly, proving that at least one of A or B is true, is the same as proving they cannot both be false. You need to learn that proving both A and B are true, is the same as proving neither one can be false.
The classic example is "All men are mortal, Socrates is a man, hence Socrates is mortal", which is actually a bit more complicated since it also involves the "quantifier" word "all". The most famous principle is "modus ponens", namely if A is true and A implies B then B is also true. (The following book may be essentially the same, but I am not positive: The book I mentioned is hard to find at a good price, but any book that explains the propositional calculus, namely what does it mean to say a given statement is true, will do. If you don't know this, you are lost when some book tries to prove that every continuous function on is bounded, and they do so by starting out assuming the function is not bounded and concluding it is not continuous. when do you know a statement is true? A fundamental fact is the "contrapositive" principle, namely that to prove that A implies B, it is entirely equivalent to prove that B being false implies A is false also. Namely what does it mean to prove something? I.e. My experience in learning to do proofs was transformed when I read the first chapter of Principles of Mathematics, by Allendoerfer and Oakley, simply because it laid out the basic facts of the "propositional calculus".
0 kommentar(er)
