 In this video, I wanna talk about the second principle of induction, sometimes referred to as strong induction, which is a variant of the first principle of induction we mentioned earlier, sometimes called weak induction, in which case, also called induction. So just reading the statement here, let Sn be a statement about integers for in as a natural number, and suppose the statement is true for Sn sub zero for some integer Sn. Also, if for all integers K, which are greater than equal to N equals zero, then S, we're gonna assume Sn sub, Sn sub zero, Sn sub zero plus one, Sn sub zero plus two, Sn sub zero plus three, up to SK, this implies SK plus one is true, then Sn is true for all integers in greater than or equal to N sub zero here. So what's the difference between the first principle of induction and the second principle of induction? The first principle of induction, your inductive hypothesis only assumes the statement is true at SK, and then using SK, you prove SK plus one. The second principle of induction, instead, we assume that all statements are true up until SK, and then that implies SK plus one. And so this is often called strong induction because the inductive hypothesis is a stronger assumption. We're assuming more things are true than as opposed to the original version of weak induction. So it's kind of weird. Weak induction just means the inductive hypothesis is weaker, which actually leads to a stronger proof, right? If you can prove more things with fewer assumptions, that's a stronger proof. So it's kind of weird. Weak induction has weaker assumptions, but it does a stronger proof. Strong induction has a stronger assumption which makes it weaker proof. So the names honestly seem a little backwards to me, but honestly, I myself don't use the names too much because honestly, strong induction is logically equivalent to weak induction. These two versions of induction are logically equivalent. That is when you're defining the actions for the natural numbers, you could take the first principle of induction and you'll can prove the second principle of induction or you can take the second principle of induction and prove the first principle. The fact that if the first principle of induction holds, then the second principle will hold very easily because the second principle takes more assumptions with the same result. So first principle implies second principle, but the domino effect actually shows that the second principle implies the first principle that the reason why you can get away with just the last assumption here is because of your base case, if you apply the inductive hypothesis to the base case, then you'll get the second case and then the third case and the fourth case all the way up to it. So even though strong induction looks like you're demanding more, it actually doesn't weaken the situation whatsoever. So you can use the first principle and the second principle interchangeably and there is no logical difference. So if you took like a introduction to proofs class like at SUU, that's math 3120. If you took a class and someone gave you a bias towards one version of induction to the other, I am telling you to throw away that bias. Basically what you're saying is, I don't like 2 fourths. 2 fourths is a dumb fraction. I only prefer 3 sixths. It seems like a very arbitrary distinction between two equivalent things. The two are interchangeable. Use whichever one is more convenient. If it's easier to prove having a stronger inductive hypothesis by all means do it. You're not losing anything. You're not a coward by using strong induction. You're actually being a very wise proof writer. So use the strong induction whenever you seem as it seems to be appropriate.