I understand how the steps in a proof are inferred from other steps that are either given or already inferred. Sometimes though, in previous lectures, a professor would begin proving a certain theorem or equation (such as in calculus and statistics) and then at some point that may as well have been arbitrary to me, declared it proven. What decides the last step in a proof and what about it is so special that it “proves” the subject?
In: 4
The last step is showing the thing that you set out to prove. It is hard to give any more info without context or an example.
If you want to learn about how to do proofs, I would suggest a proofs textbook. Mathematicians spend years learning how to write proofs elegantly and succinctly, but the journey usually begins with topics like logic and set theory. Some math majors begin their studies with a proofs class taught from a proofs textbook.
Mathematical proofs use logic as a stepping stone to get from your starting point (is this always true?) to your ending point (this is always true/false).
This used logic will depend on what you’re trying to prove, but will typically start with some type of axioms or already known truths (like arithmetic/algebra). Then it’s a matter who making the logical jump from each stepping stone until you’ve reached your ending point. That ending point will either show that your original question is always true or you can have discovered a contradiction which implies your original assumption was incorrect.
Math is simply a few concepts we have observed to be true, and we combine those simple concepts to make complex conclusions.
Imagine you found a function, where for every X you plug in, it produces a Y value that is divisible by 3.
What steps do you think you can take to say 100% for certain that all numbers it produces are divisible by 3? Sure, you might be reasonably certain but if you aren’t 100% certain, people could die if we used your formula for something like flight-navigation. That is VERY serious.
While we can’t assume all of your functions numbers are divisible by 3, we are reasonably certain that any number times 3 is divisible by 3.
We can be certain that your function multiplies x by 3, and therefore can be certain your function works bc if it isn’t then it means these rules that we as a race have observed for thousands of years to be true aren’t true.
So, instead of taking YOUR word for it, we instead take the word of every human before us and go off what they all agreed upon. If you can prove that your idea is true because if it’s false it means our understanding of math is wrong, then you have proven that either humanity is wrong or you are correct.
Many proofs are done by contradiction. Start off assuming the opposite you are trying to prove, then prove that that initial statement is impossible.
eg. an even integer plus an even integer is always an even integer.
1. m,n are even. Start off by contradiction, stating m+n=o where o is odd
2. since m, n are even, they’re divisible by 2. can be written as m=2a, n=2b
3. 2a+2b = o
4. 2(a+b) = o
5. a+b = o/2
well you started off saying m+n is an integer, a and b are integers. So o/2 must be an integer. Therefore o can’t be odd. You’ve proved m+n = even
Every step in the proof prove something. We stop when we reach the claim we originally set out to prove, but all claims that had been proved in the proof without additional assumptions are logically valid.
There are no special rule about which claim is the one we set out to prove. It’s not unusual to not have goal at first, and see what you can get, which happens a lot at research level. But when you’re in class, there is always a goal, because of 2 reasons. One, the topic is already understood, so people know which claims is important and useful, and that’s the one being taught to you. Two, if you’re being tested, then you have to be given a goal to reach.
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