Induction: Confused by trivial proof & upper and lower index

Question

I am learning induction right now and I am confused by two things.

1. This is a conceptual question, I want to prove a sigma notation using induction. $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$ (This identity doesn't not matter under this context), is it possible for n to be smaller than i? Like if i starts from 1, can n be 0 in any case?

2. I am confused by the idea of trivial proof and universal quantifiers., I saw this term in the textbook named Discrete Mathematics and Its Application. The questions goes like this:`Find a predicate such that P(1) is False, and $\forall$x$\ge$1(P(k)$\Rightarrow$P(k+1)), and k is a natural number(don't contain 0). In my understanding, I think this proposition is True, mainly because I think if P(1) is False, then this makes the implication True automatically since the hypothesis is False(This is the definition of trivial proof in the textbook described above), so if I just define a predicate like P(n)=$\frac{1}{n-n}$, this will make the hypothesis always False, hence making the implication always True at all possible natural numbers.

Do I misunderstand something here? Really need help. Thanks. :)

Answer 1

1. I see this doubt has nothing to do with induction itself but with the notation for sums, right? Any way, remeber $\Sigma_{i=0}^n a_i = a_0 + a_1 + \cdots + a_n$. In most cases, you want to evaluate that sum for $n \geq 0$. In case $n < 0 $ (let's say $n = -1$), it is generally accepted that $\Sigma_{i=0}^n a_i = 0$ by definition (as you would by summing from $a_i$ from $i = 0$ to $i = -1$, which doesn't make sense since it's not an increasing sequence).

2. This has more to do with Logic than Discrete Mathematics, indeed. This can be hard to visualize, but since can be false and actually imply some other things. I'm thinking about some trivial example for you to convince yourself, but I can't find any (i'll try later again). Anyway, note $P(n) = \frac{1}{n-n}$ is just a definition and has no logic value itself. That is, it's not true or false (although it seems to be lack of logic). Something false would be $1 < 0$ or $a\cdot 0 = a$ $\forall a \in \mathbb{R}$.

Answer 2

You misunderstood your task... You are required to provide an example of a property that is hereditary (if $P(n)$ is true then $P(n+1)$ is true) but is not universal (because $P(1)$ is not true). This shows that the hereditary is not a sufficient condition to prove a property.